REESE LIBRARY OF THE UNIVERSITY OF CALIFORNIA Received Accessions No. -^ <#. Shelf No. BARTLETT'S SPHERICAL ASTRONOMY. ELEMENTS NATURAL PHILOSOPHY BY W. H. C. BARTLETT, LL.D., PROFESSOR OF NATURAL AND EXPERIMENTAL PHILOSOPHY IN THE UNITED MILITARY ACADEMY AT WEST POINT, AUTHOR OF "ELEMENTS OF MECHANICS," "ACOUSTICS," "OPTICS," AND "ANALYTICAL MECHANICS." IV. SPHERICAL ASTRONOMY. FIFTH EDITION, REVISED AND CORRECTED. NEW YOKE: A. S. BARNES & CO., Ill & 113 WILLIAM ST., COR. OF JOHN OLD BY BOOKSELLERS GENERALLY, THROUGHOUT THE UNITED STATES. Valuable forte liy Leatii Antlers TN THE HIGHER MATHEMATICS, W. H. C. BARTLETT, LL.D., 'Prof, of Nat. Exp. Philog. in the U. ^. Military Academy, West Point* BARTLETT'S SYNTHETIC MECHANICS. Elements of Mechanics, embracing Mathematical formulae for observing and calculating the action of Forces upon Bodies the source of all physical phenomena. BARTLETT'S ANALYTICAL MECHANICS. For more advanced students than the preceding, the subjects being discussed Analytically by the aid of Calculus. BARTLETT'S ACOUSTICS ANT) OPTICS. Treating Sound and Light as disturbances of the normal Equilibrium of an analogous char- acter, and to be considered under the same general laws. BARTLETT'S ASTRONOMY*. Spherical Astronomy in its relations to Celestial Mechanics, with full applications to the current wants of Navigation, Geography, and Chronology. A. E. CHURCH, LL.D., Prof. Mathematics in the United States Military Academy, West Point. CHURCH'S ANALYTICAL G-EOMETRY. Elements of Analytical Geometry, preserving the true spirit of Analysis, and rendering 11. e whole subject attractive and easily acquired. CHURCH'S CALCULUS. Elements of the Differential and Integral Calculus, with the Calculus of Variations. CHURCH'S DESCRIPTIVE GEOMETRY. Elements of Descriptive Geometry, with its applications to Spherical Projections, Shades and Shadows, Perspective and Isometric Projections. 2 vols. ; Text and Plates respectively. EDWARD M. COURTENAY, LL.D., Late Prof. Mathematics in the University of Virginia. COURTENAY'S CALCULUS. A treatise on the Differential and Integral Calculus, and on the Calculus of Variations. CHAS. W. HACKLEY, S.T. D., Late Prof, of Mathematics and Astronomy in Columbia College. HACKLE Y'S TRIGONOMETRY. A treatise on Trigonometry, Plane and Spherical, with its application to Navigation and Surveying, Nautical and Practical Astronomy and Geodesy, with Logarithmic, Trigonomet- rical, and Nautical Tables. DAVIES & PECK, "Department of Mathematics, Columbia College. MATHEMATICAL DICTIONARY And Cyclopedia of Mathematical Science, comprising Definitions of all the terms employed in Mathematics an analysis of each branch, and of the whole as forming a single science. C H ARLES DAVIES, L L. D., Late of the United States Military Academy and of Columbia College. A. COMPLETE COURSE IN MATHEMATICS. See A. S. BABNES & Co.'s Descriptive Catalogue. Entered, according to Act of Congress, In th.e year 1859, by W. H. C. BARTLETT, In the Clerk's Office of the District Court of the United States for the Southern District of New York B'S A'MY. PREFACE. THE work here offered to the public was undertaken by ita author to supply a want long felt in his own department oi instruction in the Military Academy at West Point. Its aim is to present a concise course of Spherical Astronomy in ita relationship to Celestial Mechanics, of which it is the offspring. The solar and stellar systems are, therefore, assumed and de- scribed as necessary facts, arising from the detached condition of the bodies which compose them and the laws of universal gravitation. The consequences from these systems, to a spec- tator on the earth, are then deduced, and their entire coinci- dence with the celestial phenomena, as they arise spontane- ously, is relied upon as full and sufficient justification for the assumption, and as proof that the systems are true. This forms the first part of the subject. A general account of the methods by which the future condition and aspects of the heavens are predicted follows, and the more important appli- cations to the current wants of Navigation, Geography, and Chronology, conclude the volume. iv PREFACE. In the description and discussions of instruments, those only have been selected which are best suited to convey a full view of the whole theory and practice of Astronomical Measure- ments. The author would acknowledge his obligation to Sir John Ilerschel, Professor Challis, Mr. Maddy, Mr. Francis Bailey Mr. De Morgan, Mr. Woolhouse, M. Francceur, M. De Launay and M. Briot, whose works have been constantly before him. CONTENTS. MB Introductory Remarks. 1 Solar System S Motion < Parallactic Motion * Celestial Sphere 7 Shape of the Earth '. Diurnal Motion 9 Definitions , It Instruments 38 Proportions of Land and Water Atmosphere 13 Infraction 14 Parallelism of the Earth's Axis, and Uniformity of the Earth's Diurnal Motion.... 18 Upper and Lower Diurnal Arcs Circumpolar Bodies 18 Terrestrial Latitude and Longitude .20 Figure and Dimensions of the Earth 21 Geocentric Parallax -. 4 Augmented Horizontal Diameters 28 Distances and Dimensions of the Heavenly Bodies 29 Kcliptic . . . . 30 Precession and Nutation 87 Sidereal Time 40 Earth's Orbit 41 Mean Solar Time 4 Aberration 50 Heliocentric Parallax 53 The Seasons 58 Trade-Winds 59 Terrestrial Magnetism 62 Tides 66 Twilight 72 The Sun 78 Planets 84 Elements of the Planets 85 Dimensions and Distances of Planets . . . . VI CONTENTS. PMft Interior Planets Superior Planets 90 Synodic Revolutions Geocentric Motions 91 Direct and Retrograde Motions Stations 92 Phases of the Planets 93 Transits Occupations 95 Masses and Densities of the Planets 97 Mercury 99 Venus 1 00 Mars Planetoids 102 Jupiter Saturn 104 Uranus Neptune , 108 Secondary Bodies 109 The Moon Lunar Orbit 11G Disturbing Forces 113 Librations 115 Lunar Periods 116 Lunar Phases 117 Eclipses of the Sun and Moon 118 Moon's Relative Geocentric Orbit 123 Ecliptic Limits ; . . . 124 Number of Eclipses 125 The Saros 126 Physical Constitution oi the Moon '.27 Satellites of Jupiter 1^4 Progressive Motion of Light oi Satellites of Saturn 134 Satellites of Uranus 136 Satellites of Neptune 137 Crmets 137 Elements of the Orbits of the Permanent Comets 140 Stars 144 Elements of Stellar Orbits . 156 Proper Motion of the Stars and of the Sun 158 Nebulae , 15'J Kodiacal Light 162 Aerolites -Meteors 163 Ephemerides . . 165 Catalogue of Stars 170 Applications 174 Time of Conjunction and of Oppcsition , 174 Angle of Position 175 Projection of a Solar Ec r pse 175 Projection of a Lunar Eclipse v 183 Time of Day 184 Azimuths 1S3 Meridian Passages 191 Reduction to the Meridian 193 Terrestrial Latitude 195 Terrestrial Longitude , 208 Calendar.. .. 229 CONTEXTS AI-PEADIX I. Elements of the Principal P'.anets 23f> APPENDIX II. Astronomical Instrumc :itP 237 Clock and Chronometer 237 Vernier 243 Micrometer , 245 Level 250 Reading Microscopes 252 Transit 255 Collimating Telescope 266 Vertical Collimator 267 Collimating Eye-piece 268 Mural Circle 269 Altitude and Azimuth Instrument 276 Equatorial 282 Heliometer 293 Sextant 294 Artificial Horizon * 298 Principle of Repetition 300 Reflecting Circle 301 APPENDIX III. Atmospheric Refraction 305 APPENDIX IV. Shape and Dimensions of the Earth 31C. APPENDIX V. The Earth's Orbit 313 APPENDIX VI. Planets' Elements 316 APPENDIX VII. Planets' Elements 317 APPENDIX VIII. Planets' Elements 318 APPENDIX IX. Planets' Elements 31? APPENDIX X. Geocentric Motion 331 APPENDIX XL Mr. Woolhouse on Eclipses, &c 332 APPENDIX XIL Equation of Equal Altitudes 424 APPENDIX XIII. Correction for Difference of Refraction 42* TABLES. TABLE I. Mr. Ivory's Mean Refractions, with the Logarithms and their Differ- ences annexed .............................................. 427 TABLE II. Mr. Ivory's Refractions continued: showing the Logarithms of the corrections, on account of the state of the Thermometer and Barometer .................................................. 430 TABLE III. Mr. Ivory's Refractions continued : showing the further quantities by which the Refraction at low altitudes is to be corrected, on account of the state of the Thermometer and Barometer ....... 431 TAISI.K 1 V. For the Equation of Equal Altitudes of the Sun .................... 432 V. For the Reduction to the Meridian: showing the val.ie of _ sill V TAHLE VI. For the second part of the Reduction to the Meridian : showing the '2 sin* t P value of B = 7 ................ ...................... 450 sin I" The Greek Alphabet is here inserted to aid those who are not already amiliar with it in reading the parts of the text in which its letter occur : Letters. Names. Letters. Name*. A a Alpha N v Nu B j8e Beta S I Xi r y r CJamraa o Omicroo ^ 5 Delta n or Pi K s Epsilon V r P Rho 7 ? V Zeta 2 * Sigma H 9) Eta T T? Tau S Theta *jf u Upsilon I i Iota * Phi K * Kappa X x Chi A X Lambda Y 4, Fsi M M Mu r* <- Omega CONTENTIONAl SlGNS USED IN AsTRONOBlV. L, for mean longitude,. M, mean anomaly,. V r true anotro}y, ft, mean daily sidereal motion^ r, radius vector, p, angle of eccentpicity r <, longitude of perihelion r a, right ascension, , declination, A, logarithm of distance frona the eartliy /, heliocentric longitude, 6, heliocentric latitude, X, geocentric longitude, ft geocentric latitude, 8, longitude of ascending node f i, inclination of orbit to the ecliptic, % angular distance from perihelion to node, distance from node r or argument for latitude. ASTRONOMY, ^ UNIVEBSl ASTRONOMY 1. THE science which treats of the heavenly bodies is called Astron- omy. It is divided into Physical and Spherical Astronomy. 2. Physical Astronomy is a system of Mechanics, in which the forces are universal gravitation and inertia, and the objects the gigantic masses that move through indefinite space. It treats of the physical conditions of the heavenly bodies, their mutual actions on each other, and explains tiie causes of the celestial phenomena. 3. Spherical Astronomy is mainly concerned with the appearances, magnitudes, motions, arrangements, and distances of the heavenly bodies ; and seeks to apply the deductions from these to the practical wants of society. It is a science of observation, and its principal means of investi- gation are Optical and Mathematical Instruments. This branch of As- tronomy will form the subject of the present volume. 4. No subject calls more strongly upon the student to abandon first impressions than Astronomy. All its conclusions are in striking contra- diction to those of superficial observation, and to what appears, .at first view, the most positive evidence of the senses. 5. Every student approaches it for the first time with a firm belief that he lives on something fixed, and, abating the inequalities of hill and "alley, that this something is a flat surface of indefinite extent, composed of land and water ; and that the blue firmament which he sees around and above him in the distance is a stationary vault, upon the surface of which appear to be placed all objects out of contact with the ground. 6. The Earth on which he stands is divested by Astronomy of its flattened shape and of its character of fixidity, and is shown to be a globular body turning swiftly about its centre, and moving onward through space with great rapidity. It teaches him that his vault has no existence 2 ASTRONOMY. in fac.t, and is but a mere illusion which comes from looking through the indefinite space, extended without limit, in which he is moving. 7. Were the Earth reduced to a mere point, and a spectator placed upon it, he would see around him at one view all the bodies which make up the visible universe ; and in the absence of any means of judging of their distances from him, would refer them in the direction in which they were seen from his station, to the concave surface of an imaginary sphere, having its centre at his eye and its surface at some vast and indefinite distance. SOLAR SYSTEM. 8. A little observation would lead him to conclude that by far the greater number of these bodies appear fixed while the rest seem ever on the move, continually shifting their positions with respect to those which appear fixed, and to each other. The former are called FIXED STARS : the latter compose what is called the SOLAR SYSTEM, a group of bodies from which the fixed stars are so remote as to produce upon it no appre- ciable influence. 9. All bodies attract one another with intensities which are propor- tional to the quantity of the attracting masses directly, and to the squares of the distances inversely, Analyt. Mech., 205. 10. Bodies resist by their inertia all change in their actual state of. motion ; this resistance is exerted simultaneously with the change, and i always equal in intensity, and contrary in direction, to the force which produces it. 11. The bodies of the solar system have motions that carry them in directions oblique to the lines along which their mutual attractions are exerted. The attractive forces draw them aside from these directions ; inertia resists by an equal and contrary reaction ; and the bodies are forced into curvilinear paths, and made to revolve about the centre of inertia of the whole. 12. Thus, the antagonistic forces of gravitation and of inertia are the simple but efficient causes which keep the bodies of the solar system to- gether as a single group, and impress upon it a character of stability and perpetuity. But for the force of gravitation the bodies would separate more and more, and wander through endless space ; and but for the force ( f inertia, that of gravitation would pile them together in one confused mass. 13. The force of gravitation increases rapidly with a diminution, and decreases as rapidly with an augmentation, of distance. Those bodies which are nearest exert, therefore, the greatest influence upon one another's SOLAR SYSTEM. 3 motions. Bodies composing an insulated group may perform their evolu- tions among each other undisturbed by the action of those without, pro- vided the distances of the latter be very great in comparison to those which separate the individuals of the group. 14. This is a characteristic of the solar system. Its own dimensions, vast as they are when expressed in terms of any linear unit with which we are familiar, are utterly insignificant when compared with its distance from the fixed stars. Each of the latter, by virtue of this relatively great distance, acting upon all the bodies of the system equally and in parallel directions, the effect of the whole can only be to move the group collec- tively through space. 15. The same thing takes place upon a smaller scale within the solar system itself. Some of its members are so close together, and at the same time so far removed from the others, as to be forced to revolve about one another, while the combined action of the rest carries them as a sub-group, so to speak, about the centre of inertia of the whole. 16. The mass of the sun so far exceeds the sum of the masses of all the other bodies of the system, as to throw the centre of ineuia of the whole group within the boundary of its own volume ; and although the centre of the sun actually revolves about this point, yet its motion bo- tomes so small, when viewed from the distance of the earth, that it is in- sensible except through the medium of the most refined instruments. All the other bodies are, therefore, said to revolve about the sun as a centre, and it is from this fact, and the controlling influence which this latter body exerts over the motions of all the others, that the system takes its name. 1 7. The same is true of the sub-groups ; the mass of one of the bodies in each being so much greater than the sum of the masses of the rest as to cause the latter to revolve approximately about its centre, while this centre revolves about the sun. 18. The path a body describes about another as a principal source of attraction, is called an orbit. 19. Those bodies which describe their orbits about the stn aie called primary, and those which describe their orbits about the primaries are called secondary bodies. These latter are also called Satellites. Of the primary bodies there are three distinct classes, differing from each other mainly in the shape of their orbits, their densities, and gen- eral aspects. 20. A body subjected to the action of a central force, whose intensity varies as the square of the distance inversely, mut describe one or other of 4 ASTRONOMY f he conic sections, depending upon the relation Between its velocity and the intensity of the central force. The orbits that are known to belong to the solar system are ellipses. 21. Those primaries which move in elliptical orbits of small eccentri- cities are called PLANETS. Those primaries having orbits of great eccentri- cities are called COMETS. Comets are also distinguished from planets in having a degree of density so low as to give some the appearance more of a vapor than of a solid body. 22. The solar system consists then of the Sun, Planets, Comets, and Satellites. Setting out from the sun, the known planets, with their names, occur in the following order, viz. : Mercury, Venus, the Earth, Mars, then a class called the Planetoids, of which ninety-one are known at the present time, Jupiter, Saturn, Uranus, and Neptune. See Plate I., Fig. 1. To these must be added a multitude of much smaller bodies of the nature of planetoids, whose existence is inferred from the fact that some of their number make their way now and then to the earth's surface under tfie name of meteors. 23. It would be utterly impossible to give within the narrow limits of an octavo page a graphical representation of the relative dimensions of the solar system ; and to aid the conceptions of the student, Sir John Herschel has instituted the following illustration, viz. : On any well-levelled field place a globe two feet in diameter ; this will represent the sun ; Mercury will be represented by a grain of mustard-seed on the circumference of a circle 164 feet in diameter for its orbit ; Venus a pea on the circumference of a circle 284 feet in diameter ; the Earth also a pea on the circumference of a circle 430 feet in diameter; Mars a rather large pin's head on the circumference of a circle of 654 feet diameter ; the Planetoids grains of sand on circular orbits varying from 1000 to 1200 feet in diameter; Jupiter a moderate sized orange on a circumference nearly half a mile in diameter ; Saturn a small orange on the circumference of a circle four- fifths of a mile in diameter ; Uranus a full sized cherry on the circumfer- ence of a circle more than a mile and a half in diameter ; and Neptune a good sized plum on the circumference of a circle about two miles and a half in diameter. To illustrate the relative motions, Mercury must describe a portion of its orbit equal in length to its own diameter in 41 seconds; Venus in 4 minutes and 14 seconds ; the Earth in 7 minutes ; Mars in 4 minutes and 48 seconds ; Jupiter in 2 hours and 56 minutes ; Saturn in 3 hours and 13 minutes; Uranus in 2 hours and 16 minutes, and Neptune in 3 hours and 30 minutes. Now conceive the two feet globe to be in- creased till its diameter becomes 880,000 English miles, and suppose the Plate I. TO FttOATT PAW 4. SOLAR SYSTEM. 5 other bodies and their distances increased in the same proportion ; the re- sult will represent the dimensions of the solar system. It will give to the earth a diameter of nearly eight thousand miles, a distance from the sun equal to 95 millions of miles, and a velocity through space, around the sun, of 19 miles a second. The orbits, although referred to as circles, are in fact ellipses, but of ec- centricities so small as to justify the substitution for the mere purposes of the illustration. 24. The fixed stars are self-luminous. The sun is regarded as one of this class of bodies, and by its greater proximity to the earth, becomes the principal source of heat and light to its inhabitants. 25. The planets and satellites are opaque non-luminous bodies, and are visible only in consequence of light received from the sun and reflected to the earth* SPHERICAL ASTRONOMY. MOTION. 26. Motion signifies the condition of a body, in virtue of which it oc cupies successively different places. But we can form -no idea of place ex- cept by referring it to other places, and these again, to be known, must be referred to others, and so without limit ; so that place is, in its very nature, entirely relative. Motion is, from its definition, therefore, also relative. 27. We judge of the rate of motion by the greater or less rapidity with which the object possessing it varies its distance from other objects assumed as origins. These origins may themselves be in motion, but if the circumstances of the spectator be such as to deceive him into the belief that they are at rest, he will attribute all change of distance to a motion wholly in the object which he refers to them. And this is one of the most fruitful sources of the many erroneous notions with which students gener- erally commence the study of astronomy. 28. If two objects be in motion, and they alone occupy the spectator's field of view, the effect to him will be the same if he suppose one fixed, and attribute the whole of its motion to the other in a contrary direction ; for this will not alter the rate by which they approach to or recede from one another. PARALLACTIC MOTION AND PARALLAX. 29. The real motion of a spectator gives rise to the appearance of motion among surrounding objects which are relatively at rest. Objects in front of him seem to separate from one another, those behind appear to approach one another, and those directly to the right and left seem to move in a direction parallel to his own motion. A spectator, for example, travelling over a plain studded with trees or other objects will, on fixing his eyes upon a single object without with- drawing his attention from the general landscape, see or think he sees the CELESTIAL SPHERE. y latter in rotary motion about that object as a centre ; all objects between it and himself appearing to move backward, or contrary to his own motion, and all beyond it, forward or in the direction in which he moves. This apparent change in the relative places of objects, arising from a shifting of the point of view from which they are seen, is called parallactic motion ; and the amount of angular change in the instance of any partic- ular object is called the parallax of that object. 30. Let P be the place of an object, C and S the places from which it is seen ; and let its place be referred to some point Z', on the prolongation of the line CS, which joins the points of view. The angular change in the place of P as seen from C and S will be Z' SP-Z' CP=SP C= the parallax of P. That is to say, the parallax of an object is the angle sub- tended at the object by the distance between the stations from which it is seen. Make CP=d; <7S=p; the angle Z' SP=Z; the angle SPC=z Then from the triangle C S P, we have sin 2= ^ . sin Z d Whence the parallax increases with an increase of the spectator's change of place, with diminution of the object's distance, and also with the approx imation of Z to 90. 31. All other things being equal, the parallax will be less as the ob- ject's distance is greater ; and when the parallax is zero for any arbitrary value of Z, the factor - must be zero, and the change of the spectator's place must be utterly insignificant in comparison with the object's distance CELESTIAL SPHERE. 32. Now, when the heavens are examined it is found that by far the greater number of the celestial bodies have no sensible parallax, while comparatively a few have. The first are the fixed stars ; and they are so called from the fact that they always preserve the same angular distances from any assumed point and from each other, from whatever station on the earth they are viewed. The second are bodies of the solar system. 33. The fixed stars are, therefore, beyond limits at which objects cease 8 SPHERICAL ASTRONOMY to be sensibly affected by parallax. The great concave of the heavens upon which the fixed stars appear to be situated, is called the celestial sphere. Not only, therefore, is the longest rectilineal dimension of the earth, but also the distance between the points of its orbit about the sun most remote from each other a distance, as we shall see in the sequel, equal to one hundred and ninety millions of miles utterly insignificant when expressed in terms of the radius of the celestial sphere as unity. A sphere large enough to contain the entire orbit of the earth is a mere point in comparison with the vast volume embraced by the celestial sphere. The centre of ike earth may, therefore, always be regarded as the centre of the celestial sphere. Fig. 2. SHAPE OF THE EARTH. 34. The earth, being the station from which all the other heavenly bodies are viewed, is the first to claim attention. It has been repeatedly circumnavigated in dif- erent directions, and the portions of its sur- face visible from elevated positions in the midst of extended plains or at sea, always appear as circles of which the spectator seems to occupy the centre. The apparent diameters of these circles, measured by in- struments, are smaller in proportion as the points of view S are more elevated. The earth is, therefore, ylobular ; for to such figures alone belong the property of always presenting to the view a circular outline. 35. By the figure of the earth is meant its general shape without regard to the irregularities of surface which form its hills and valleys. These are relatively insignificant and are disregarded in speaking of the earth's form. They are less in proportion to the entire earth than the protuberances and indentations on the surface of a smooth orange are te a large size specimen of that fruit. The earth is an oblate spheroid, and the operations and method of computations by which its precise magni tude and proportions are found, will be given presently. The shortest diameter of the earth is called its axis. DIURNAL MOTION. 9 DIURNAL MOTION. 36. The boundary of the visible portion of the earth's surface, sup posed perfectly smooth, is called the sensible horizon. The sensible horizon Fig. 3. Fig. 4. is only seen at sea, or on extended plains. At most localities on land it is broken by hills, valleys, and other objects. 37. The earth conceals from us that portion of space below our sen- sible horizon, while all above is exposed to view. It rotates upon its axis, and the period required to perform one entire revolution is called a day. 38. Every spectator is carried about the earth's axis in the circumference of a circle, and while the extent of the visible portion of space remains un- changed, different regions are continu- ally passing through the field of view. The horizon of a spectator will be ever depressing itself below those bodies which lie in the region of space towards which he is carried by the rotation, and elevating itself above those in the oppo- site quarter ; thus successively bringing into view the former and hiding the latter. 39. The spectator being unconscious of his own motion, concludes, from first appearances, that his horizon is at rest, and attributes these changes to an actual motion in the objects themselves. Instead of his horizon approaching the bodies, he judges the bodies to approach his horizon ; and when it passes and hides them, he regards them as having sunk below it or set, while those it has just disclosed, and from which it is receding, he considers as having come up or risen. 40. One entire revolution about the axis being completed, the spec- tator returns to the place from which he commenced his observations, and he begins again to witness the same succession of phenomena and in the same order. All the heavenly bodies appear to occupy the same places m the concave sky which they did before. 41. Thus the rotation of the earth about its axis produces the daily 10 SPHERICAL ASTRONOMY rising and setting of the sun the alternation of day and night ; also the rising and setting of the other heavenly bodies, their progress through the vault of the heavens, and their return to the same apparent places at short *nd definite intervals. 42. The apparent motions with reference to the horizon by which these daily recurring phenomena are brought about, are called the diurnal motions of the heavenly bodies. The real motion is in the horizon, the origin of reference ; it is only apparent in the bodies themselves. DEFINITIONS. 43. The axis of the celestial sphere is the axis of the earth produced. 44. The poles of the earth are the points in which its axis pierces its surface. The pole nearest to Greenland is called the north, the other the south pole. 45. The poles of the heavens are the points in which its axis pierces the celestial sphere. That above the north pole of the earth is called the nor th, the other the south pole. 46. The earth's equator is the intersection of the earth's surface by a plane through its centre, and perpendicular to its axis. . 47. The equinoctial is the intersection of the surface of the celestial sphere by the same plane. 48. A meridian line is the intersection of the earth's surface by a plane through its axis and the place of a spectator. 49. The celestial meridian is the intersection of the surface of the celestial sphere by the same plane. This is often called simply the me- ridian of the place. 50. The poles of the celestial meridian are called the East and West points ; that towards which the spectator is moving by his diurnal motion being the East, that from which he is receding the West. 51. The apparent zenith and apparent nadir are the points in which a plumb-line produced intersects the celestial sphere : that over head being the zenith. 52. The rational horizon is the intersection of the celestial sphere by a plane through the earth's centre and perpendicular to the line of the zenith and nadir. The plumb-line being always normal to the earth's sur- face, the plane of the rational horizon is parallel to the plane tangent to the earth's surface at the spectator's place, and these- planes intersect the celestial sphere sensibly in the same great circle. 53. The dip of the horizon is the angle which the elements of a DEFINITIONS. jj visual cone, whose vertex is in the eye of the spectator, and whose surface is tangent to that of the earth along the sensible horizon, make with the tangent plane to the earth at the spectator's place. The dip is greater in proportion as the spectator's elevation above the earth is greater. When the eye is in the earth's surface, the dip is zero, and the visual cone be- comes the tangent plane. This coincidence will always be supposed to exist unless the contrary is specially noticed. 54. The latitude of a place on the earth's surface is the arc of the celestial meridian from the equinoctial to the zenith of the place. It is always measured in degrees, minutes, seconds, and thirds. Latitude is reckoned north or south ; that reckoned towards the north pole being called north latitude, that towards the south pole, south latitude. The greatest latitude a place can have is 90, this being the latitude of the poles of the earth. 55. Parallels of latitude are small circles on the earth's surface par- allel to the equator. All places on the same parallel have the same latitude. 56. The longitude of a place on the earth's surface is the arc of the equinoctial intercepted between the meridian of the place and that of some otlfer place assumed as a first meridian. It is called East or West, according as it is reckoned in the direction from the first meridian towards its east or west point. For the sake of uniformity, it will, in the text y al ways be reckoned in the latter direction. The English estimate longitude from the meridian of Greenwich, the French from that of Paris, and other nations from other meridians. In the United States, for most geographical purposes, it is estimated from the meridian of Washington. 57. A vertical circle is the intersection of the celestial sphere by a plane through the zenith and nadir. The prime vertical is the vertical circle whose plane is perpendicular to that of the meridian. 58. The north and south points are the poles of the prime vertical ; that below the north pole being called the north point. 59. The Azimuth of a body is the angle which a vertical circle through the body's centre makes with the meridian. It is measured on the horizon, and from the south towards the west, or from the north to- wards the west, according as the north or south pole is elevated above th horizon. It may vary from to 360. 60. The zenith distance of an objec is the angular distance from the apparent zenith to the centre of the object, measured on a vertical circle. 12 SPHERICAL ASTRONOMY. 61. The altitude of an object is the angular distance from the horizon to the object's centre, measured on a vertical circle. The azimuth and zenith distance are a species of polar co-ordinates for the designating an object's place in the heavens. By making the azimuth vary from zero to 360, and the zenith distance from zero to 90, every visible point of celestial space may be defined in position. 62. A declination circle, or hour circle, is the intersection of a plane through the axis of the heavens with the celestial sphere. 63. The declination of an object is the angular distance of its centre from the equinoctial, measured on a declination circle. The declination may be north or south, and may vary from to 90. 64. The polar distance of an object is the angular distance of its centre from the celestial pole, measured on a declination circle. 65. The right ascension of an object is the angle which a declination circle through the object's centre makes with a declination circle through a certain point on the equinoctial, called the Vernal Equinox. This angle is measured upon the equinoctial, and eastwardly in direction. 66. The polar distance and right ascension are also a kind of polai co-ordinates for defining the places of celestial objects ; for this purpose it is only necessary to cause the right ascension to vary from to 360, and the polar distance to vary from to 180, to reach every point in the celestial sphere. 67. The hour angle of an object is the angle which its hour circta makes with the meridian of the place. It is estimated from the meridian westwardly, and may vary from to 360. The hour angle may be em- ployed, instead of the right ascension, with the polar distance to define an object's place. To illustrate, let the plane of the paper be that of the meridian ; the circle HZ ON its intersection with the celestial sphere; P P' the axis of the heavens ; P and P' the north and south poles respectively ; Z and N the zenith and nadir^ respectively, and the earth a mere point at (7; then will the circle QWQ'E, of which P and P' are the poles, be the equinoctial; HWOE, of which Z and N are the poles, the hori- zon ; E and W, the poles of the meridian, will be the east and west points respectively; the arc ZQ will be the latitude, ZSA a vertical circle, INSTRUMENTS. 13 Z S the zenith distance of the object S, A S its altitude, and WA its azimuth ; PS will be its polar distance, D S its declination, Z P S, meas- ured by Q />, its hour angle, and if V be the vernal equinox, V D will be its right ascension. INSTRUMENTS. 68. Most of the data with which the practical astronomer labors, come from measurements made in the circles just referred to, by means of certain astronomical instruments. These instruments are described, and their theory, adjustments, and uses explained, in Appendix II. The student should study, in connection with short daily lessons of the text, from this point, the Clock. Chronometer, Transit, Mural Circle and Azimuth and Altitude Instrument. The others should be taken up where referred to, in the order of the text. PROPORTIONS OF LAND AND WATER. THE ATMOSPHERE. 69. To resume the consideration of the earth. About three fourths of its surface are covered with water, and the greatest depth of the sea does not probably exceed the greatest elevation of the continents. The earth is surrounded by a gaseous envelope, called the atmosphere, the actual thickness of which, were it- reduced to a uniform density throughout, equal to that at the surface of the sea, would be about five miles. But owing to the law which regulates the pressure, density, and temperature of elastic bodies, it is much greater than this. The dif- ferent strata, being relieved from the weight of those below them, become more expanded in proportion as they are higher, and the place of the su perior atmospheric limit must result from an equilibrium between the. weight of the terminal stratum and the elastic force of that upon which it rests. The laws just referred to indicate that this limit cannot be much higher than 80 miles. 70. The atmosphere is not perfectly transparent. The sun illumines its particles ; these scatter by reflection the light they receive, particularly the blue, in all directions, and produce that general illumination called daylight and gives to the sky its bluish aspect. But for this diffusive power of the air, no object could be visible out of direct sunshine ; the shadow of every passing cloud would be pitchy darkness, the stars would Lo visible all day, and every apartment into which the sun did not throw his direct rays would .be involved in total obscurity. In ascending to SPHERICAL ASTRONOMY. the summits of high mountains, the diffused light becomes less and less, the sky deepens in hue, and finally, at great altitudes, approaches to total blackness. 71. The superior illumination of the atmosphere produced by the solar light obliterates, as it were by contrast, the light from almost all the other heavenly bodies, and few, if any, of the latter are seen when the sun is up. REFRACTION. 72. Luminous waves which enter the atmosphere obliquely are, ac- cording to the laws of optics, deviated by the latter from their course, and made to exhibit the objects from which they proceed in positions different from those they actually occupy, and thus false impressions are produced in regard to true places of the heavenly bodies. Take, for example, a spectator fig. 86. on the earth at A ; and let L D L represent a section of the supe- rior limit of the atmosphere, and KA A' that of the earth's sur- face by a vertical plane. A star at S would, in the absence of the atmosphere, appear in the direction A S ; but in reality, when the portion of the luminous wave moving on this line reaches the point J9, it is turned down- ward, and made to come to the earth at some point A', pursuing a course such as to bring its suc- cessive positions normal to some curve, as DA', whose curvature increases towards the earth's surface, in consequence of the increasing density of the atmosphere in that direction. This part of the wave cannot therefore go to the spectator. Not so, how- ever, with a portion of the same general wave incident at some point as D\ nearer to the zenith ; this, after pursuing a path D'A similar to DA', will reach the spectator at A, and cause the body from which it originally proceeded to appear in the direction A S', tangent to the curve at the point A, the effect being the same as though the body had shifted its place towards the zenith by the angular distance S A S'. 73. The air's refraction, therefore, diminishes apparently the zenith REFRACTION. 13 distances of all bodies, and increases their altitudes. Any body actually in the horizon will appear above it, and any body apparently in the horizon must be below it. 74. It is also obvious that refraction can only take place in the ver- tical plane through the body, since this plane is always normal to the surfaces of the atmospheric strata, and divides them symmetrically. Re- fraction will not, therefore, in general, affect the azimuth of a body. 75. This apparent angular displacement of a body from its true place, caused by the action of the atmosphere upon its luminous waves, is called refraction j and various formulas have been constructed to compute its exact amount. One of the best of these is by Littrow, which has the merit of depending upon no special hypothesis in regard to the constitu- tion of the atmosphere, being constructed upon the most general prinoi pies, and from known and well-ascertained data. 76. Make, Z = Z f A S' = observed zenith distance ; r = S A S 1 = corresponding refraction ; h = height of mercurial column, which the atmosphere supports ; t = temperature of the air and of the mercury ; a coefficient of atmospheric expansion for each degree of Fahr. ; (3 = coefficient of expansion for mercury, same thermometric scale Then, Appendix No. III., / 9.4-si'n 2 ^V (2) '30'l + (*-50)a ' or, omitting the last term in the parenthesis as being insignificant for or dinary zenith distances, r = 57".82. ,- -^. tan Z .(I -0.0012517 sec'Z) . . (3) 30 1 + (t 50ja When h = 30, and t = 50, equation ( 3 ) becomes r in = 57".82 tan Z (1 - 0.0012517 sec 2 Z) = A . . (4) and the results given by this formula for different values for Z are called mean refractions ; and for any other state of the thermometer and barometer. h 1 + (50 *)/3 r = A 'W'l + (t-5V)*> and taking logarithms, , . A 1 + (50-0# , K \ logr=::log^ + log- + log r ^- . . . (5) 16 SPHERICAL ASTRONOMY. Causing Z to vary from to 90, h from 28 to 31 inches, and t from 80 to 20, the logarithms above may be computed arid tabulated for future use, under the heads Z, t, and b. 77. Causing Z to vary from to 90, in equation (4), we may construct Table I.; causing t to vary from 80 to 20, and h to vary from 31 to 28, in the last two terms of equation (5), we may construct Table II. Returning to equation (2), resuming the quantity omitted to obtain equation (3), computing their values for zenith distances, varying from 75 to 90, on the supposition that A=30 and =50, an additional table may be computed to correct the refractions in low altitudes. Tables L, II.. and III. are due to Mr. Ivory. 78. For zenith distances exceeding 80, refraction becomes very uncertain ; it then no longer depends solely upon the state of the atmo- sphere, which is indicated by the barometer and thermometer, being fre- quently found to vary at the same station some 3 to 4 minutes for the same indications of these instruments. Example. The zenith distance of an object is observed to be 71 26' 00", the barometer standing at 29.76 in., and the thermometer at 43 Fahr : required the refraction. Table I. Mean refraction, log. 2.23609 Table II. Barometer 29.76 " 9.99651 Table II. Thermometer 43 " 0.00668 Hefraction 2' 53".49 . . 2.23928 Observed zenith distance . 71 26' 00".00 Zenith dist. cleared from refraction 71 28' 53 ".49 The refraction must always be added to the observed zenith distance, or subtracted from the observed altitude, to clear an observation from re- fraction. PARALLELISM OF THE EARTH'S AXIS, AND UNIFORMITY OF THE EARTH'S DIURNAL MOTION. 79. Wherever upon the earth's surface the altitudes and instru- mental azimuths of a star are taken in the various points of its diurnal course, and the instrument is turned in azimuth, so as to read the half sum of two azimuths, corresponding to any two equal altitudes, the vertical plane through the line of collimation is found to divide the path symmet- PARALLELISM OF THE EARTH'S AXIS. 17 rically ; and this plane of symmetry for any one star will, at the same place of observation, also be a plane of symmetry for all the stars. In other words, the diurnal paths of the stars may be divided symmetrically by any number of planes inclined to one another through the earth's centre a condition which can only be fulfilled for paths upon the celes- tial sphere, when these paths are circles, of which the poles coincide,, and; the planes qf symmetry pass through them. The diurnal motions of the stars are only apparent, and arise from an, actual motion of the spectator about the earth's axis. This latter line preserves, therefore, its direction unchanged, and, in the motion of the earth around the sun, describes a cylindrical surface, of which the elements have their vanishing point in the poles of the celestial sphere. These poles are therefore the geometric poles of the diurnal paths of the stars, and the planes of symmetry are the meridian planes of the places of < >bservation. 80. Again, the interval of time during which a star is moving be tween any two given altitudes on one side of the plane of symmetry, is exactly equal to that during which it is moving between the equal alti- tudes on the opposite side, which can only be true, for all positions of the observer, when the star's apparent, or the earths real motion about its axis, is uniform. 81. The period of one revolution of the earth about its axis is called H day; the day is divided into 24 equal parts called hours; the hours into 60 equal parts called minutes ; the minutes into 60 equal parts called seconds, and the seconds into 60 equal parts called thirds. 82. The earth rotates therefore at the rate of 360^-24 = 15 an hour ; 15' of space in 1 minute of time ; 15" of space in 1 second of tiro^, or 15'" of space in 1 third of time. 83. Distances on the equinoctial may therefore be expressed in time or space at pleasure, the former being convertible into the latter by multiplying by 15, or the latter into the former by dividing by 15. 84. To distinguish hours, minutes, and seconds in time, from degrees, minutes, and seconds in arc, the formerare usuall) designated by the nota tion h, m, s, and the latter by , ', " ; thus an arc upon the equinoctial may be written 357 39' 38", or 23 h 50 m 38 8 .5. 85. To find the instrumental azimuth of the meridian of a place. Bring the line of collimation of an altitude and azimuth instrument, prop- erly levelled, upon a star in the east or west, clamp the vertical circle, and read the instrumental azimuth ; then by an azimuthal motion bring the line of collimation upon the star when in the west or east, and again read the 2 18 SPHERICAL ASTRONOMY. azimuth : the half sum of the two will be the instrumental azimuth sought. To bring the line of collimation into the meridian, turn the instrument till it reads this half sum. UPPER AND LOWER DIURNAL ARCS. CIRCUMPOLAR BODIES. Fig. 87. 86. The diurnal paths of the heavenly bodies which are cut by the horizon are, in general, divided by the latter unequally. The portions of these paths above the horizon are called the upper, and those below the lower diurnal arcs. 87. To find, for any spectator, thb relation which these arcs bear to one an- other, let PQP'Q' be the meridian, P the elevated pole, Q Q' the equinoctial, Z the zenith, H W H' the horizon, S'SS"S" r the diurnal path of any body, the earth being a mere point at E ; then will S r S S" be the upper, and S" S" f S 1 the lower diurnal arc. Make / = Q Z, latitude of the spectator, p = P S", polar distance of the body, P = ZP S", the hour angle of the body when in the horizon, 2 = Z S", zenith distance of the body in horizon. Then in the triangle ZP S", because PZ -90 J, cos z = cos p sin I + sin p cos I cos P . . . . (6) but 2 = 90, whence == cos p sin I + sin p cos I cos P ; or cos P = - tan I If I = 0, or p = 90, then will cos P = 0, and P 90 = 6 h ; that is, if the spectator be upon the equator, or the body upon the equi- noctial, the semi-upper arc will be six hours, and the body will be a* long above as below the horizon.- CIRCUMPOLAR BODIES. 19 If p < J, then will cos P < 1 ; which is impossible, and the place of the body can never satisfy the con- dition that z = 90. In other words, when the jolar distance is less than the latitude of the spectator's place, the body can never sink to the horizon, and will ever remain in the field of perpetual apparition. Such bodies, as well as their diurnal paths, are said to be circumpolar. If p /, then will cosP 1; P=180=12 h ; that is, when the polar distance of the body is equal to the latitude of the spectator's place, the body can never sink below the horizon, but will iust gra^ze it in the meridian. If p > /, and p < 90, cos P < 0, cos P > 1 ; P > 90, P > 6 h ; that is, all bodies between the elevated pole and the equinoctial, will be longer above than below the horizon. If p > /, and p > 90, cos P > 0, cos P < 1, P < 90, P < 6 h ; that is, if the body and the spectator be on opposite sides of the plane ol the equinoctial, the semi-upper arc will be less than six hours, and the body will be a shorter time above than below the horizon. If p = 180 Z, then will tan p = tan /, and cos P= 1, P=0 = h ; that is, when the body is at a distance from the depressed pole equal to the latitude of the place, the body will never rise above the horizon, but just graze it in the meridian. If p > 180 I, then will tan p > - tan /, and cos P > 1, which is impossible. That is to say, if the body's distance from the de- pressed pole be less than the spectator's latitude, the body can never ris to the horizon, and must ever remain invisible. 88. The act of a body's passing the meridian, is called its culmina tion. A body has its greatest or least altitude at the instant of its cul mination. The altitude of a body when on the meridian is called its meridian altitude. SSO SPHERICAL ASTRONOMY, TERRESTRIAL LATITUDE AND LONGITUDE. 89. Latitude. When in Eq. ( 7 ) the angle then will p = I ; but in this case p is the polar distance of the point of the horizon of the same name as the elevated pole, and hence the latitude of the spectator is always equal to the altitude of the elevated pole. 90. This suggests an easy and accurate method of getting from obser- vation both the latitude of the specta- tor's place and the polar distance of a star. Let Z be the zenith, HIT the hori- zon, Q Q' the equinoctial, P the eleva- ted and P' the depressed pole, and S' S, the diurnal path of a circumpolar star. Make / = HP = Z Q , the latitude, p = P S' = P S , the polar distance of star, a' H S f , the greatest observed meridian altitude of star, a { = HS , the least observed meridian altitude of star, r' and r t , the refractions corresponding to the greatest and least meridian altitudes respectively. Then from the figure will ^.i^a=i.ii^EaJ ... ( 8) P = That is to say, the latitude of the observer's place is equal to the half sum of the greatest and least meridian altitudes of a circumpolar star ; and the polar distance of the star is equal to the half difference of its greatest and least meridian altitudes. Other methods for finding the latitude will be given in another place. 91. Longitude. The uniform motion* of the earth about its axis ftir- nishes the means of finding the longitude of the spectator's place. Twenty-four perfect time-keepers, with dial-plates graduated to 24 hours, placed upon meridians 15 apart, and so regulated as to mark 24 h at the instant any one fixed star or other point of the heavens culminates, would, FIGURE OF THE EARTH. 21 82, when this regulating star or point comes to any one of these me- ridians, simultaneously mark the hours indicated by the natural numbers from one to twenty-four, inclusive; that 15 to the east of the regulating point marking l h , that 30 to the east marking 2 h , and so on to that 345 to the east, or 1 5 to the west, marking 23 h . The timepieces to the east would be later and later, those to the west earlier and earlier. The times indicated on these several timepieces are called the local times of their re- spective meridians. 92. If now, without altering its hands or rate of motion, a traveller were to transport the time-keeper of any one of these meridians to that on any other, and note the difference of time indicated by the two, this differ- ence would be the difference of longitude of the two meridians, expressed in time ; and multiplied by 15 would give the same in degrees. 93. If one of these meridians be the first meridian, this difference would be the longitude of the other. But if neither be the first meridian, this difference applied to the longitude of one, supposed known, would give the longitude of the other. 94. The solution of the problem of longitude consists, therefore, in finding the difference of the local times which exist simultaneously on the first and required meridians. The various modes of doing this will be given in another place. FIGURE AND DIMENSIONS OF THE EARTH. 95. A fluid mass rotating about an axis, and of which the particles attract one another with intensities varying inversely as the square of their distances apart, will assume the form of an oblate spheroid. Its axis of rotation will be both the shortest and a principal axis of figure. Where the angular velocity is such as to make the centrifugal force of the sur- face elements small in comparison with their weight, due to the attraction of the whole mass, the figure of the meridian section will, ( 265, Analyt. Mechanics,) approach that of an ellipse of small eccentricity. 96. The centrifugal force of a body at the equator of the earth, where it is greatest, is only about ?j-f th part of its weight. Observations upon the temperature of the strata composing the earth's crust, lead to the conclusion that at no great depth below its surface its materials are in a fluid state from excessive heat ; and the researches of geology make it more than probable that there was a time when the earth was without Solid matter. Its present irregularities of surface, forming mountains, hills, valleys, the bed of the ocean, of seas, lakes and rivers, are due to 22 SPHERICAL ASTRONOMY. changes subsequent to the surface induration from cooling, and as the ver- tical dimensions of these are insignificant in comparison with the depth to the centre of the entire mass, it is concluded that the figure of the earth is one of fluid equilibrium due to its rotary motion. 97. Assuming the meridian section of the earth to be an ellipse, its eccentricity and semi-axes are found, Appendix No. IV., from the relations c c' ' c sin 2 1' c' sin 8 1-V (10) (H) (12) Fig. in which e = the eccentricity of the meridian ; A = semi-transverse axis = equatorial radius of the earth ; B = semi- conjugate axis = polar radius of the earth ; c and c' = the linear dimensions of the arcs of the meridian, whose extremities differ in latitude by 1 ; l m and l' m latitudes of the middle joints of the arcs c and c' respec- tively. The quantities l m , l' m , c, c r are found from observation and measure- ment. A method by which / and /' may be found is explained in 90. 98. To find c and e', a base line AB is carefully measured on some extended plain, and a number of stations (7, D, E, F, H, ., -A**./. and by subtraction />, /, = />, P III I.,, = Q. II D I , I ; that is, the radius of the zone of greatest polar illumination, or obscuration, is equal to .the greatest declination of the sun. 133. Two small circles parallel to the equinoctial, and at a distance from the poles equal to the greatest declination of the sun, are called polar circles ; that about the north pole is called the arctic, and that about the south the antarctic circle. The polar circles are the boundaries of the greatest zones of polar diurnal illumination and obscuration. 134. When the intervals of time between three consecutive passages of a circumpolar star over the line of collimation of a transit or mural cir- cle are equal, these instruments are adjusted to the meridian. 135. The diurnal motion brings the meridian of a place, in the course of one revolution of the earth on its axis, into coincidence with the decli- nation circle of every body in the heavens. The difference of times between the meridian's passing the centres of any two bodies, is the difference of right ascension of these bodies. 136. To find the time o* the meridian's passing the centre of any body, find by the transit instrument and timepiece the time of th<: merid- ian's passing the body's east and west limb, and take half the sum. 3 34. SPHERICAL ASTRONOMY. 137. To find the polar distance of a body's centre, take the reading of the mural circle when its line of collirnation is upon the upper or lower lirnb ; subtract from this the polar reading and correct the difference for refraction, parallax in altitude, and semi-diameter. The declination is ob- tained by subtracting the polar distance from 90. 138. The points in which the equinoctial intersects the ecliptic are called the equinoxes ; that by which the sun passes from the south to the north of the equinoctial is called the vernal equinox ; the other, or that by which the sun passes from the north to the south of the equinoctial, is called the autumnal equinox. 139. The angle which the equinoctial makes with the ecliptic is called the obliquity of the ecliptic. 140. To find the place of the vernal equi- nox and the obliquity of the ecliptic, let VD Z be an arc of the equinoctial, VS 2 of the eclip- tic, V the vernal equinox, $, and S z two places of the sun when on the meridian at different times, ,/>,, S Z D 2 arcs of declina- tion circles ; and make <, = D { $ b the sun's declination at any meridian passage ; # 2 = D 2 S& the same at some subsequent passage ; 2a = VD 2 VD { , the corresponding difference of right ascension ; x = VDi, the right ascension of the sun at the time of first meri 2 , right-angled at D { and D 2l sin x tan #, cot w, sin (x -f- 2a) = tan tana. ... (31) sm ( 2 t ) Also cot w = sin x . cot <5, (32) The value of the obliquity is thus found to be nearly 23 27' 54", which is therefore the greatest north and south declination of the sun. The tropics are, therefore, 23 27' 54" from the equinoctial, and the polar circles are at the same distance from the poles. 141. The interval of time between the sun and a star crossing the meridian, applied to the right ascension of the sun, gives the right ascen- :ion of the star. The declination of a star is found like that of the sun, except that there is no correction for parallax and semi-diameter, the only correction being for refraction. 142. The right ascension and declination of one star being known, the differences of observed right ascensions and declinations, the latter being corrected for differences of refractions, give, when applied to the right as- cension and declination of the known star, the right ascension and decli- nation of other stars. Thus a list of the stars, together with their right ascensions and declinations, and arranged in the order of their right ascen- sions, furnishes the ground-work of what is called a catalogue of stars, of vvhich a fuller account will be given presently. 143. A belt of the heavens extending on either side of the ecliptic, far enough to embrace the paths of the planets, is called the zodiac. 144. The ecliptic is divided into twelve equal parts, called signs. They commence at the vernal equinox, and are named in order, proceed- ing towards the east, Aries (T), Taurus (), Gemini (n), Cancer (?), Leo (ty), Virgo ("HE), Libra (===), Scorpio (^), Sagittarius ( # ), Capricor- nus (V?), Aquarius (~), and Pisces (X). Motion in the order of the signs is said to be direct ; the converse, retrograde. 145. The points of the ecliptic in which the sun reaches his greatest 'north and south declination are called the solstitial points : that on the north is called the summer solstice, and that on the south the winter sol- stice. The sun when in these points appears to be stationary as regards his apparent motion in declination. The solstitial colure is the declination circle through the solstitial points. The equinoctial colure is the declina- tion circle through the equinoctial points. The solstitial colure separates Gemini from Cancer, and Sagittarius from Capricornus; the equinoctial colure separates Aries from Pisces, and Virgo from Libra. 146. A great circle of the celestial sphere passing through the pole> of the ecliptic is called a circle of latitude. SPHERICAL ASTRONOMY. g 147. The latitude of a body is the distance of the bouy's AH tie from the ecliptic, measured on a circle of latitude. 148. The longitude of a body is the distance from the vernal equinox to the circle of latitude through the body's centre, measured on the eclip- tic in the order of the signs. The longitude and latitude are co-ordinates that refer a body's place to the circle of latitude through the vernal equinox and to the ecliptic ; the longitude and ecliptic polar distance are polar co-ordinates that refer a body's place to the same circle of latitude and to the pole of the ecliptic- 149. The longitude of the sun, as seen from the earth, is readily ob- tained from the obliquity of the ecliptic and either the right ascension or declination. For this purpose make Fi - u bis - a =: VS^ the longitude of the sun; 8 SiDn his declination; a = VDi, his right ascension ; w = S { VD h the obliquity of the ecliptic. Then will tan a = sin = tan a COS W sin 5 (33) (34) 150. The place of the sun as seen from the earth, and that of the earth as seen from the sun, are at the opposite extremities of the same di- ameter of the ecliptic; and the longitude of the sun, increased by 180, will be the longitude of the earth as viewed from the sun, the centre of the earth's orbital motion. 151. The sun appears in the vernal equinox on the 20th March, in the autumnal equinox on the 22d September, the summer solstice on the 21st June, and in the winter solstice on the 21st December. The poles of the ecliptic are at a distance from the nearest poles of the equinoctial, equal to the obliquity of the ecliptic. g 152. The right ascension is obtained from observation by means of the clock and transit instrument, the declination by means of the mural circle. From these and the obliquity of the ecliptic, the longitude and latitude are obtained from computation. Thus, let S be the body's place, V the vernal equinox, VD the body's right ascension, D S its declina- PRECESSION AND NUTATION. tion, VL its longitude, S L its latitude, and the *"* angle L VD the obliquity of the ecliptic. Make // \ a = VD = right ascension ; ^z^/L \ \ \ X = L S = latitude ; w = L VD = obliquity of the ecliptic ; e = S V = arc of great circle through the body and vernal equinox ; 9 = S VD = inclination of S V to the equinoctial Then in the triangle S VD, right-angled at D, tan 5 ten 9 = (35) sin a cos e = cos a . cos (36) and in the triangle VL S, right-angled at Z, tan I = tan e . cos (9 00) (37) sin X = sin s . sin (9 w) (38) 153. If the longitude, latitude, and obliquity be given, then in the triangle VLS, tan X tan (9 w) = -r-j (39) cos e = cos / . cos X (40) and in the triangle VD S, tan a = tan s . cos 9 (41) sin 5 = sin s . sin 9 ...... (42) PRECESSION AND NUTATION. 154. The longitudes and latitudes of the stars, being thus determined at different epochs, show a slow increase in all the longitudes, while the latitudes remain sensibly the same. 155. This is owing to a slow gyratory motion of the line of the ter- restrial poles in a retrograde direction, caused by the rotary motion of the earth and the combined action of the sun, moon, and planets upon the ring of equatorial matter that projects beyond the sphere of which the polar axis is the diameter. It is the resultant of two component motions. SPHERICAL ASTRONOMY. Fig. 4a 156. By the first of these components alone, called nutation, the line of the poles would describe once in every 19 years an acute conical surface, of which the vertex is at the centre of the earth, and the intersec- tions with the celestial sphere are two equal ellipses, whose transverse and conjugate axes are respectively 18". 5 and 13".74, the former being al- ways directed towards the poles of the ecliptic 157. By the second, called the mean precession, the centres of these ellipses are carried uniformly around the poles of the ecliptic from east to west in equal circles, of which the radii are about 23 28', and at a rate of 50 ".2 in the interval of time between two consecutive returns of the sun to the mean vernal equinox. This interval is called a tropical year. The mean equinoxes perform, therefore, one entire revolution in 360 4- 50".2 = 25817 tropical years. In the figure, S is the sun, E the earth, P t the north pole of the ecliptic, P' the true and P the mean north pole of the equinoctial. The eurva about S represents the earth's orbit, that about E, and of which the plane is perpen- dicular to E P', shows the direction of the earth's axial motion ; E V is the intersection of the plane of this circle with that of the ecliptic, and Fis the vernal" equinox. The circle about P, has a radius of about 23 28', the curve about P is the elliptical path de- scribed by the true pole P f about the mean P, and of which the longer axis P'P" passes through P r The arc V L of the ecliptic, the circle about P,, and ellipses about P are all on the surface of the celestial sphere, while them, are at its centre. 158. By the component motions of mean precession and of nutation combined, the true equinoctial pole is. carried with a variable motion along a gently waving curve whose undulations extend to equal distances on either side of the circumference of mean precession, and which it intersects at points separated by angular distances, as seen from the centre of the celestial sphere, equal to 19 X 50".2 X sin 23 28'-=- 2 13",74 = 3' 10" 18",74. 159. The motions due to the action of the sun afld moon are opposed to those arising from the ac- tion of the planets, and when estimated along the E, and the curves about Fig. 47. PRECESSION AND NUTATION. $Q ecliptic are called luni-solar precession in longitude. The combined effect aii sing from the simultaneous action of all the bodies, estimated in the same direction, is called the general precession in longitude. 160. The equinoxes always conforming to the places of the equinoc- tial poles, have a slow, irregular, but continuous retrograde motion. The place of the vernal equinox without nutation is called the mean, equinox ; with nutation, the true or apparent equinox. The inclination of the equinoctial to the ecliptic without nutation is called the mean obliquity ; with nutation, the true or apparent obliquity. The difference between the mean and apparent obliquity is called the nu- tation of obliquity. 161. The apparent equinox wanders in either direction from the mean to a distance equal to 13", 74-^2 sin 23 28',= 17", 25, which it reaches when the mean and apparent obliquity are equal ; and the apparent obliquity varies on either side of the mean from zero to half of 18 x/ .5 or 9 /x .25; the latter being reached when the apparent equinox coincides with the mean. 162. The motion of the mean equinox along the ecliptic is deter- mined by that of the centre of the little ellipse above referred to, and is therefore at the rate of 50".2 a year, being the quotient which results from dividing 360 by the period required for the true pole to perform one entire circuit around the pole of the ecliptic. 163. The distance from the mean to the apparent equinox is called the equation of the equinoxes in longitude. 164. The intersection of a declination circle through the mean equi- nox with the equinoctial, is called the reduced place of the mean equinox. 165. The distance from the reduced place of the mean equinox to the apparent equinox, is called the equation of the equinoxes in right as- cension. 166. The changes which take place in these equations, as also in the apparent obliquity of the ecliptic, are called periodical variations, from the circumstance of their running through all their possible values in a com- paratively short period. Formulas for computing the equations of the equinoxes in longitude and right ascension will be given in another place. 167. Besides the motion of the equinoctial, due to the action of the heavenly bodies on the protuberant ring of matter about the terrestrial equator, there is another effect due to the deflecting action of the planets-. By this the earth is turned aside from the path it would describe, if subject- ed to the action of the sun alone, and the place of the ecliptic, therefore, 7 40 SPHERICAL ASTRONOMF. changed. The amount of this change is exceedingly smal , being only about 46" in a century. Its present effect is to diminish the mean obli- quity, and this will continue to be the case for a long period of ages, when the change will be in the opposite direction, the motion being one of oscil- lation to the extent of 1 21' about a mean position. The change in the value of the mean obliquity arising from the cause here referred to, is called the secular variation of the obliquity, because of the great period of time required to pass through all its values. SIDEREAL TIME. 168. It has been explained (p. 237) how the motion of the pointers or hands of clocks and watches over stationary circular scales of equal parts upon their dial-plates, is employed to measure the lapse of time. The uni- form motion of the meridian, carrying with it an imaginary movable circu- lar scale of equal parts, coincident with ^he equinoctial, gives the means of regulating these and all other artificial time-keepers. 169. The origin or zero of the equinoctial scale is on the upper me- ridian; its unit of measure is one hour, equal to 15 ; its pointer or hatid the declination circle through the centre of some heavenly body,' and time measured upon it takes the name of the body which regulates the pointer. 170. The distance of the pointer from the origin or upper meridian, estimated westwardly, is the hour angle of the body which gives the scale its name, and measures the time since its meridian passage. 171. Time measured by the hour angle of the mean equinox is called mean sidereal time ; and the interval of time between two consecu- tive passages of the meridian over the mean equinox, is called a sidereal day. 172. Time measured by the hour angle of the apparent equinox is called apparent sidereal time ; and the interval of time between two con- secutive passages of the meridian over the apparent equinox, is called an apparent sidereal day. 173. Apparent sidereal time is that usually employed by astronomers. It is affected by the equation of the equinoxes in right ascension, of which the value in time being applied to the apparent sidereal time, with its proper sign, gives the mean sidereal time. This difference between appz. rent and mean sidereal time is called also the equation of sidereal time. 174. Apparent sidereal days are slightly unequal; but the fluctua- tions of a clock marking apparent from one noting mean sidereal time would be only about 2*.3 in nineteen years. 175. A timepiece whose hour-hand passes unif>rmly over the circular THE EARTH'S ORBIT 41 scale of 24 hours on the dial -plate, in a sidereal day, is said to run with sidereal time ; it will mark mean sidereal time when its hands indicate at any and every instant the hour angle of the mean vernal equinox. 176. The sidereal time of the meridian's passing the centre of any body is the true right ascension of the body; and the rate of the time- piece on sidereal time, its error at any epoch, and the indication of the hands on its dial-plate at the instant the meridian passes the centre of any body, are the data which make known the body's right ascension. 177. The sidereal day is shorter than the time required for the earth to turn once about its axis by about T J^ of a sidereal second. THE EARTH'S ORBIT. 178. The orbit of the earth is an ellipse, of which the sun occupies one of the foci. 179. The extremities of the transverse axis of the orbit are called the Apsides ; that most remote from the sun is called the higher and that nearest to the sun the lower apsis. The lower apsis is also called the pe- rihelion and the higher apsis the aphelion. The transverse axis produced both ways is called the line of the apsides. 180. The place of the sun or other heavenly body which has the greatest distance from the earth is called the apogee, and that which has the least distance is called the perigee. When, therefore, the earth is in aphelion, the sun is in apogee ; and when the earth is in perihelion, the sun is in perigee. 181. The quotient obtained from dividing the circumference of a circle, of which the radius is unity, by the interval of time between two consecutive returns of a body to the same origin, is called the body's mean motion from that origin. Thus, let T be the interval, and m the mean motion ; then will 182. The origin may be movable or fixed ; when in motion, the mo- tion may be direct or retrograde. 183. Denote by r the radius vector of the earth, by^c the area which this line describes in a unit of time, and by n the true motion, tl en will, Analytical Mechanics, equation (266), ,= 2 ......... (44) SPHERICAL ASTRONOMY. 184. The interval of time between two consecutive returns of the sun to the vernal equinox, is called a tropical year. That between two consecutive returns to the mean vernal equinox, a mean tropical year. 185. The arc of the ecliptic from the mean vernal equinox to the place the sun would occupy, had his motion in longitude been uniform and equal to a mean of his actual motions, is called his mean longi- tude the true and mean places always coming together on the line of the apsides 186. The interval of time between two consecutive returns of the earth to the perihelion or aphelion is called an anomalistic year. 187. The mean motion of the earth from perihelion is the value of m, in equation (43), the value of T therein being the anomalistic year. 188. The angle E S P, which the radius vector of the earth makes at any time with the line of the apsides, reckoned from perihelion, is called the true anomaly. 189. The angle which the radius vector of the earth at any time would make with the same line, and estimated from the same point, had the earth moved from perihelion with its mean motion, and retained this motion unaltered, is called the mean anomaly. 190. The relation which connects the mean with the true anomaly 'is, Appendix No. V. ? equation (g), n = V 2 e sin V -f- f e 9 sin 2 F &c. . . . (45) in which n is the i jean anomaly, V the true anomaly, and e the eccen- tricity. 191. The difference between the mean and the true anomaly is called the equation of the centre. Denoting the equation of the centre by E, we have, equation (45), E = n V = 2 e sin V + f e 2 sin 2 V &c. . . (46) 192. Let 'S, S. S a S w represent the ecliptic ; S v S a the line of the equinoxes ; S s S w the line of the solstices ; S v the vernal equinox ; S the sun ; PEA E a P the earth's orbit ; P the perihelion ; A the aphelion. When the earth is at E v the sun will appear at the vernal equinox $; when at E t , the sun will appear at the summer solstice S, \ and when at E a . the sun will THE EARTH'S ORBIT. 43 appear at the autumnal equinox S a ; and when at E w , the sun will appear at the winter solstice S v . 193. Let E t be the place of the earth, E m its mean place ; then will S v E' t , estimated in the order of the signs, that is, in the direction indi- cated in the figure, be the earth's longitude as seen from the sun ; S v E' m , estimated in the same direction, its mean longitude ; S v P' the longitude of the perihelion ; P'E' t the true anomaly ; P'E' m its mean anomaly, and E' t S E' m the equation of the centre. 194. It is obvious that the equation of the centre is equal to the dif- ference between the mean and true longitudes from the same equinox. 195. The earth's orbit is known when its semi-transverse axis, its ec- centricity, and the longitude of its perihelion are known, its plane being that of the ecliptic. These are called the elements of figure. The periodic time, the mean motion, and the mean longitude at some particular epoch, are the additional data from which result by computation the earth's true motion and actual place at any other epoch before or after. These are called the elements of place and motion. 196. Make L = mean longitude of the earth at the given epoch ; t = an interval of time before or after ; m = mean motion ; a = true longitude at time of observation ; a p = longitude of the perihelion : then, Appendix No. V., equation (i), L + m t = a 2 e sin (a a p ) + f e* sin 2 (a a^) &c. (47) The sun will have the greatest apparent diameter when the earth is in perihelion, and least when in aphelion ; denote these diameters by ; and d f respectively, and the corresponding radii vectors by r t and r' ; then from the principles of optics, and r'-r,:r'+ ,-,::$,-*': whence r'-r,_ S-t- r'+r,~ -*, Actual measurements give about 5, = 32',5 ' = 31 '.5 44 SPHERICAL ASTRONOMY. whence e = = 0.016 nearly ; from which it appears that e is so small as to justify the omission from equation (47) of those terms in which its powers higher than the first enter, and we may write L + m t = a 2 e sin (a a, f ) (48) 197. From four observed right ascensions of the sun, compute, by equation (33), his corresponding true longitudes ; each longitude increased by 180 will give the corresponding true longitude of the earth ; denote these by a,, a^ a 3 , and a 4 , and the intervals of time from the epoch of the mean longitude Z, say noon, January 1st, to the times of observation, by *,, 4, ^ 3 , and t 4 respectively, then will, equation (48), L + m t { = a, 2 e sin (a, a p ), ^j L + m t, = a z 2 e sin (a, a,), I L + m t 3 = a 3 2 e sin (a 3 a,), j ' L -f- m t 4 = a 4 2 e sin (a 4 a^), J four equations, from which the mean longitude L at the epoch, the mean motion m, the eccentricity e, and longitude of the perihelion a p , may be found. For this purpose subtract the first from the second, m ('* $,) = a 8 a, 2 e [sin (a 2 a p ) sin (a, a,,)] ; making t s ti = , . (54) * = 2 sin i a sin i a" . sin (" ) * ) we have Mn + Ni + k.M.N= 0. Replacing M and JVby their values given in equations (51), we find n (m & a) i(m& a) k (m 6 of _ ^ md'-a f h m6"a" (m &' ~'"" ~ whence (m6a) [(nb" + i &' + k &) m- (n a" -f ia' -f- a)] = 0. But m & a cannot be zero, since e is not zero. 46 SPHERICAL ASTRONOMY. Placing, therefore, the second factor equal to zero, we find a" n a." + ^ a' + *# a w, w / =- = < 55) From equations (54) we have 1 _ _ ^i_?^i 2 ( a " - a ) } n sin ^ a' . sin ^ (a' a)' k_ sin la", sin |(a".- a') f ..... n sin * a . sin i- (a' a) ' J Now a, a', and a'' are the increments of the true longitude since the i k first observation; these in equations (56) give the fractions - and -; these in equation (55) give the value of m\ this in equations (51) and (52) give the value of tan (a { a p ) and therefore of a p ; this, in equation (50), gives the value of e, and this, together with m and a p , in first of equations (49), gives the value of L. 198. The mean motion in longitude, the eccentricity and longitude of the perihelion being determined at dates remote from one another, are found to be very slightly variable. The present value of the eccentricity is 0.01678356, the semi-transverse axis of the earth's orbit, or the earth's mean distance from the sun being unity ; that of the mean motion in lon- gitude in one sidereal day is 0.98295603 ; the longitude of the perigee at the beginning of the present century was 279 30' 05 ".0, and the mean longitude of the sun at the same time was 280 39' 10'''. 2. The longitude of the perihelion is found to increase at a mean rate of 61 ".9, in a tropical year, and deducting 50".2 for the retrocession of the mean equinox, gives to the perihelion a direct motion of 11 ".7 through space in the same time. 199. Denoting by y^ the length of the tropical year in sidereal days, we have 0.98295603 = * v ' MEAN SOLAR TIME. 200. Although the mode of reckoning time by the motion of the ver- nal equinox affords great facilities in practical astronomy, it is of little or no use in the ordinary operations of common life. Business and social in- MEAN SOLAR TIME. 47 tercourse are mostly regulated by the alternations of daylight and darkness, and the sun is the natural object of reference in all divisions of time for so- ciety in general. 201. Time measured by the hour angle of the sun is apparent solar time. 202. The epoch of the sun's being on the meridian of a place, is called apparent noon of that place. 203. The interval of time between two consecutive passages of the sun's centre over the upper or lower meridian of the same place, is called an apparent solar day. The apparent solar is longer than the sidereal day, in consequence of the eartli's real, and therefore of the sun's apparent, motion in the ecliptic in an easterly direction. If, for instance, the vernal equinox and the sun 'were to pass the meridian of a place at the same instant to-day, the sun would be to the east of the equinox on' the morrow, and would cross the same meridian after it. 204. The orbital motion of the earth and, therefore, the apparent mo- tion of the sun in the ecliptic is, Eq. (a), Appendix V, variable. The unequal arcs which measure the daily increments of the sun's longitude vary their inclination to the equinoctial from about 23 28' at the equi- noxes, to zero at the solstices ; and these unequal arcs may hence be pro- jected by declination circles into still more unequal arcs cf right ascension. These latter measure the excess of the different apparent solar over the si- dereal days ; and hence the variable orbital motion of the earth, and the in- clination of the plane of its orbit to the equinoctial, conspire to make the lengths of the apparent solar days unequal. 205. Timepieces cannot be made to imitate this inequality, nor is it desirable they should do so, were it possible. Had the earth's orbit been circular and in the plane of the equinoctial, its orbital motion would have been uniform, the sun's apparent daily in- crease of right ascension constant, and the apparent solar days of equal duration. 206. These conditions are fulfilled by the device of an imaginary sun conceived to move uniformly in the equinoctial with the true sun's mean motion in longitude, and to set out from the reduced place of the mean vernal equinox when the true sun's mean place leaves the mean equinox. This imaginary body is called the mean sun. 207. Time measured by the hour angle of the mean sun is called mean solar time. The epoch of the mean sun being on the meridian of a place, is called mean noon of that place. SPHERICAL ASTRONOMY. Fig. 49. 208. The difference between the apparent and mean -solar time is called the equation of time. If to the mean time the equation of time be applied with its proper sign, the apparent time will result ; if the equation of time be applied with its proper sign to the apparent time, the mean time will result. The equation of time is employed to pass from mean to apparent, or from apparent to mean time. 209. Thus, let P M be an arc of the meridian, VM of the equinoctial, VE of the eclipuc ; P the pole of the equinoctial ; V the true, V m the mean, and F r the reduced place of the mean equinox ; S the true and S m the mean sun ; then will MP S be apparent, and MP S m mean solar time ; VS a the right ascension of the real sun : V V r the equation of the equinoxes in right ascension. Make e S a S m = the equation of time ; a = VS a = the right ascension of true sun ; I = V T S m = the mean longitude of the sun ; q =. W r = the equation of the equinoxes in right ascension j then from the figure, we have e = a-(l + q) (58) that is, the equation of time is equal to-the sun's true right ascension di- minished by the sun's mean longitude, corrected for the equation of the equinoxes in right ascension. 210. When the sun's true right ascension exceeds the corrected mean longitude, the equation of time must be added to apparent time to obtain mean time, and vice versa. The equation of time is zero four times a year, viz., on 15th April, 14th June, 31st August, and 24th December. 211. The mean sun and mean equinox when together must pnss some meridian at the same instant. .When the same meridian returns to the mean equinox on the following day, the mean sun will be to the east by a distance equal to that which measures its motion in one sidereal day ; and the mean solar day will exceed the sidereal day by the interval of sidereal time required for the meridian to overtake. the mean sun after it passes the mean equinox. Denote this excess by , expressed in days ; and the motion of the mean MEAN SOLAR TIME. 49 sun in one sidereal day, equal to the earth's mean orbital motion in the same time, by m. Then will m t be the motion of the mean sun in the time t, and its right ascension from the mean equinox at the instant the meridian overtakes it will be m + m t. But this is the hour angle of the mean equinox, or the sidereal time t, reduced to degrees ; whence m + mt= 360 X t', or t= m 360- m' and for the length of the mean solar day, expressed in sidereal time, ^ m + 360 m' or replacing m by its value 0.98295603, 198, and denoting the length of the mean solar day by D m , expressed in terms of the sidereal day D,, as unity, we have D m = 1.00273791 D (59) and Whence to convert intervals of mean solar into intervals of sidereal, or in- tervals of sidereal into intervals of mean solar time, we have these rules, viz. : Sidereal interval = 1.00273791 X Solar interval, Solar interval = 0.99726957 X Sidereal interval. 212. Applying this second rule to the length of the tropical year ex- pressed in sidereal days, we have, Eq. (59), Solar interval = 0.99726957 X 366.242 = 3&5.2422414 ; or reducing the fraction to hours, minutes, 'and seconds, and denoting the length of the tropical year, expressed in mean solar time, by y tm , we have y tm = 365 d 5 h 48^ 48 s ....... (60) 213. Denote by y am the length of the anomalistic year expressed in mean solar time; then, 157 and 198, 360 - 50".2 : 360 -f- 11". 7 : : 365 d 5 h 48" 48* : y am ; whence y= 365 d 6 h 13 m .3 (61) 214. The interval of time required for the earth to perfcrm one entire circuit about the sun in space is called a sidereal year. 50 SPHERICAL ASTRONOMY. Denote by y tm the length of the sidereal year in mean solar time, then 360 50".2 : 360 : : 365 d 5" 48 m 48 s : y. m ; whence y m = 365 d 6 h 9 m 9 8 .6 (62) ABERRATION. 215. The earth's orbital motion, combined with the motion of light, produces an apparent displacement of all the heavenly bodies in the di- rection of the point of the celestial sphere towards which the earth is, at the instant, moving. This displacement is called aberration. Thus, let S be the place of a heavenly Fig. 50. body, E that of the earth moving from M towards N along an arc of its orbit. From E take any distance E E' ; join S and .", and lay off upon E' S the distance E' C, which bears to EE', the ratio of the velocity of light to that of the specta- tor, and suppose C connected like himself with the earth. Now a wave of light from S, another which originates at (7, when that from S E E' passes this point, and the spectator's eye starting from E at the same instant, will all meet at E\ and as bodies al- ways appear in the direction ot the normal to the wave front, the point C and the body S will be seen in the direction E' S. But (7, having a ve- locity equal and parallel to the spectator, will have passed on to C", at the extremity of a line through (7 equal and parallel to E E' ; so that when C appears in the direction of the body S, it will, in fact, be in advance of it by the angle S E' C'. Let C be the optical centre of the object-glass of a telescope, attached to the face of a graduated circle, moving in the plane of the body and the tangent line to the terrestrial orbit at the earth's place, and E' the inter- section of the cross wires at the solar focus ; then, when the image of the body appears at the latter point, the line of collimation will be in advance of the body itself, and its instrumental bearing will be in error by the angle S E' C', and must be corrected by the same angle to get the true bearing. 216. But had 5 been a terrestrial object, by the t'nie its light frcto ABERRATION. 51 the position S had reached (7, the body itself would have been at $", the intersection of E C produced and S S' drawn parallel to E E' ; and at the instant of its light reaching E' the body would have been at *S", the intersection of S S'' produced and the line of collimation. Geodetical observations are, therefore, unaffected by aberration, while astronomical observations are, in general, affected by it. 217. Make = aberration ; a = S E' N = angle the direction of the body makes with that of the earth's motion. V = velocity of the earth ; V = velocity of light : Then, in the triangle G,E' E, V : V : : sin (a r) : sin r, whence y sin r = . sin (a r) (64) If p, denote the mean radius of the earth's orbit, then will 2*> t 365 d .25636 ' and it will be shown hereafter that light requires 16 m 26' to pass over the distance 2 p,, and therefore whence r_*Hi*xir*r V 365 d .25636 from which, and equation (64), it is apparent that r is very small, and may be neglected in comparison with a ; we may therefore write 0.00000815 sin a, 206264".8 in which 206264.8 is the number of seconds in radius ; whence r"= 0.00009815 X 206264".8 sin a, or r"= 20".246 sin a (66) 52 SPHERICAL ASTRONOMY. 218. Let A B be the intersection of the celestial Fig. 51. sphere by a plane through the body and the direction of the earth's motion, A C that of a plane through the observer and star, and perpendicular to the plane of the ecliptic, and B C an arc of the ecliptic ; then will B be the point in which the tangent to the earth's orbit at the place of the earth pierces the celestial sphere, A will be the projection of the body upon the celestial sphere, and A B = a ; and if A C = X and C A B = 9, we have cos 9 = tan X cot a, and cos 2 9 = tan* X . cot 2 a, whence sm 2

u * t _ make with the line of the zenith and nadir to the earth's surface, an angle equal to BAC, equal to the sun's zenith distance, which being denoted by z, we have , A C = A B . sec z. THE SEASONS. 57 Denote by /' the temperature when the wave and earth surfaces are par- allel, and by / when they are oblique ; then AB . secz : AE \\ I' \ /; whence != = !' .cos z; sec z and if I t denote the temperature which would result at the unit's distance from the sun, and r the radius vector of the earth, we have from the law of diffusion, depending upon distance, whence /=--. cos z ........ (73) 235. Resuming Eq. ( 6 ), and making p = 90 o?, in which d de- notes the sun's declination, we have cos z = sin I . sin d + cos / . cos d . cos P . . . (74) which, in Eq. (73), gives I ~* - [ sm ^ sin c? + cos / . cos d . cos P] . . (75) This result is wholly independent of terrestrial longitude, and is only de- pendent on the latitude of the place, the sun's declination, and the place of the earth in its orbit. All places upon the same parallel are equally exposed, therefore, to the solar influence, and whatever differences of mean temperature and of climate they may exhibit are due to local causes, such as the vicinity of mountains, extended plains, forests, deserts, or large bodies of water, upon all of which the sun is known to produce great va- riety of thermal effects. 236. Making 2 = 90, in Eq. (74), we have cos P = tan / . tan d ...... (76) and making P = 0, in Eq. (75), we have I=^cos(l-d) ....... (77) Eq. (76) gives the value of the semi-upper diurnal arc, or the time the sun is above the horizon, or the duration of calorific action ; and Eq. (77) thr< intensity of the solar influence when greatest. 58 SPHERICAL ASTROI OMY. 237. In the course of the tropical year the declination varies nearly 47, the sun beiug at one time about 23.5 north, and at another about the same distance south of the equator. As long as the latitude and declination are of the same name, that is, both north or both south, the sun will, Eq. (76), be longer than twelve hours above the horizon, and the place will receive more heat than it loses. And in proportion as the latitude and declination approach to equality, the intensity of the solar action will, Eq. (77), approach its maximum. This periodical variation in the daily average temperature of a place, caused by a change of the sun's declination, gives rise to the phenomena of the seasons. 238. The interval of time during which the daily increment of tem- perature of a place is increasing is called its spring ; that during which this increment is decreasing is called its summer ; that during which the daily decrement is increasing is called its autumn or fall ; and that during which this decrement is decreasing is called its winter. 239. Within the tropics C ' C' and D D', and especially about the equator Q Q', the temperature is, Eqs. (76) and (77), nearly uniform, and always high. On this account the terrestrial belt bounded by the tropics is called the torrid zone. Between the tropics and polar cir- cles A A' and BE' the average daily temperature is much less uniform and always lower than in the torrid zone. The belts bounded by the tropics and polar circles are called temperate zones. Between the poles P and P' and polar circles, the variation of the av- erage daily temperature is the greatest possible and the temperature itsell least. The portions of the earth's surface about the poles and bounded by the polar circles are called frigid zones. 240. Places within the torrid zone may be said to have two of each of the seasons during a tropical year, and all places in the temperate and frigid zones but one. For all places in the north temperate and frigid zones, spring begins when the sun is on the equator and passing from south to north, or on the 20th March ; summer, when the sun reaches the tropic of Cancer, or on the 21st June; autumn, when the sun returns to the equator in passing to TRADE WINDS. 59 the south, or 22d feeptember; and winter, when the sun reaches the tropic of Capricorn, or 21st December. For all places in the south temperate and frigid zones the names of the seasons will be reversed spring becomes autumn, and summer winter. 241. The elliptic form of the earth's orbit causes the radius vector, and therefore, Eq. (77), the intensity of the solar heat, to vary. But the angular velocity of the earth about the sun also varies, and according to the same law, viz. : that of the inverse square of the earth's distance from the sun Analytical Mechanics, Eq. (266). Equal amounts of heat will therefore be developed while the earth is describing equal arcs of longitude, and the supply will be the same during the description of any two seg- ments, equal or unequal, into which the entire orbit is divided by a line through the sun. The earth is nearer the sun while the latter is south of the equinoctial, or from the latter part of September to the latter part of March ; and it describes the corresponding part of its orbit in a time so much shortened as just to balance the increase of thermal intensity. But for this law of compensation, the effect would be to increase the difference of summer and winter temperature in the southern and to diminish it in the northern hemisphere. As it is, however, no such inequality is found to subsist, but an equal and impartial distribution of heat and light is ac- corded to both hemispheres. 242. But it must not be inferred that the mean surface heat is con- stant throughout the year; for such is not the fact. By taking, at all sea- sons, the mean of the temperatures of places diametrically opposite to one another, Professor Dove finds the mean temperature of the whole earth's surface in June considerably greater than that in December. This is due to the greater amount of land in that hemisphere which has its summer solstice in June ; the thermal effect of the sun on land being greater than that on water. 243. The variation of the radius vector amounts to about ^ of its mean value, and therefore the fluctuation of heat intensity to about j 1 ^ of its average measure a circumstance which is manifested in a great excess of local heat in the interior of Australia during a southern, over that of the deserts of Africa during a northern summer. TRADE WINDS. 244. A discussion of the trade winds, the earth's magnetism, and the tides, belongs, in strictness, rather to terrestrial physics than to astronomy ; but the accessary connection of these phenomena with the earth's diurnal 6u SPHERICAL ASTRONOMY. rotation and the action of foreign bodies upon the earth, as wll as their importance to navigation, make a sufficient apology for introducing them here. 245. The surface of the torrid zone is most heated ; its excess of temperature is communicated to the superincumbent atmosphere; the latter is expanded, and becoming specifically lighter, is pressed upward by the Bolder portions on the north and south which move in and take its place. These, in their turn, are heated, expanded, and pressed upward, and a constantly ascending current is thus produced ovor an entire zone, of which the boundaries fluctuate with the varying declination of the sun and the proportion of land and water on the belt of the earth's crust lying immediately under the sun's diurnal path. The air thus accumu- lated at the summit of the ascending column, being unsupported on the north and south, flows oft' under the action of its own weight in either di- rection towards the poles, and, after cooling, descends again to the earth's surface in the higher latitudes of the temperate zones to supply the place nnd follow the course of that which has passed to the torrid zone. 246. Two atmospheric rings, as it were, distinguished by peculiarities of internal circulation, are thus made to belt the earth on either side of the equator in directions paral- lel or nearly so to that great circle. On the lower side of these rings, in contact with the earth, the air moves towards the base of the ascending col- umn, and on the upper towards the poles. 247. By the diurnal mo- tion of the earth, places on the equator have the greatest velo- city of rotation, and all other places less in the proportion of the radii of their respective parallels of latitude. The portions of the ascending column which flow towards the poles set out with the east- ward intertropical velocity, which they carry with them in part to the higher latitudes, where they descend to the earth's surface. To an ob- server situated in these latitudes, the air will have an apparent east- wardly motion, approaching to the excess of the intertropical velocity over that of the observer's parallel. Here westerly winds prevail. 248. On parallels a few degrees lower, the tendency of the air is TEADE WINDS (JJ towards the equator, and this combined with what remains of the apparent easterly component, just referred to, gives rise in the north- ern hemisphere to a northwesterly and in the southern to a southwesterly wind. 249. In its onward course towards the equator, this same air crosses successively parallels of greater and greater velocity, and this, together with friction against the earth's surface, reduces the air's excess of easterly motion to zero, and here northerly winds prevail in the northern and southerly winds in the southern hemisphere. 250. In latitu les still lower, the excess of rotation is in favor of the earth's surface, and the air, unable to keep up, now lags behind, and ap- parently tends to the west ; and here, if the places be in the northern hemisphere, northeasterly, and if in the southern hemisphere southeasterly winds prevail. 251. Nearer to the equator the radii of the parallels vary less rap- idly, and the velocities of places on the same meridian are more nearly equal. In crossing these parallels the air in its onward course finds less variation in the velocity of the earth's surface, and friction, which now urges the air to the east, together with the easterly pressure below, arising from the westerly lagging in the summit of the ascending column, due to its decreasing angular motion as it recedes from the centre of rotation, soon brings the air and earth to relative rest. This occurs within the base of the ascending column where the currents of air, which are continually approaching each other from the directions of the poles, meet. This is, therefore, a region of calms. 252. The aerial currents thus produced under the combined influence of solar 'heat and the diurnal motion of the earth, are called Trade winds ; and they are so called from the benefits they are continually conferring on trade dependent upon navigation. 253. A voyage from the United States to northern Europe in a sailing vessel is on an average ten days shorter than in the contrary direc- tion. A sailing vessel on a passage from northern Europe to the southern coast of the United States would proceed to the Madeiras to take the east- erly trades, and returning would proceed to the Bermudas to catch west- erly trades. 254. Within the region of calms the ascending column of air car- ries with it a large amount of aqueous vapor. In its ascent the air expands, its temperature is depressed, its aqueous vapor is first condensed into clouds, then into rain, and thus the region of calms is also a region of dense clouds and copous rains ; the former giving to the earth, as viewed from 62 SPHERICAL ASTRONOMY. a distance, the appearance of being girted tiy dark broken belts, arranged in zones parallel to the equator. 255. The limits of the trades do not always occur in the same lati- tudes, but vary with the season. In December and January, when the sun is furthest south, the northern boundary of the northeast trades of the Atlantic is about 20 N., whilst in the opposite season, from June to Sep- tember, it is 32 N. 256. Owing to the great disparity in the effects of solar heat upon land and water, and to the influence of mountain ranges and valleys upon atmospheric currents, the regular trades only occur, as a general rule, at sea, though in some level countries, within or near the tropics, constant easterly winds prevail. This is remarkably the case over the vast plains drained by the Amazon and lower Orinoco. 257. The trades of the ocean and of the land are separated by a belt, within which other and variable winds occur. This belt lies upon the ocean, and extends along the coasts. When to the east of the trades, it is often a hundred miles wide, but when to the west its width is much smaller. The interruption of the trades, here referred to, is due to the difference of temperature of the air on sea and land, which changes with the seasons. The air over the land in the higher latitudes is the warmer when the meridian zenith distance of the sun is least, and colder when greatest. During the first period the wind is from the sea to the land, and in the second from the land to the sea, thus giving rise to the period- ical winds called Monsoom, which occur even within the limits of the trades. A large island thus circumstanced is surrounded by a wind blow- ing from all quarters at the same time. 258. A similar difference of temperature, but which varies with the alternations of day and night, gives rise to what are called the sea and land breezes. TERRESTRIAL MAGNETISM. 259. Another most important effect from the solar heat, combined with the diurnal motion of the earth, is the earth's magnetism. 260. A difference of temperature in different parts of any body form- ing a continuous circuit is ever accompanied by electrical waves, propa- gated from the hotter to the colder parts. If the circuit be composed of various materials, possessing different powers of conducting heat, this differ- ence may be maintained in greater degree and duration, and the effects of the electrical flow rendered more strikingly manifest. TERRESTRIAL MAGNETISM. ^3 261. When the source of heat is moved gradually along the circuit, the electrical flow is in the direction of this motion, the colder portions always lying in advance and the warmer behind the moving source. 262. A compass-needle, brought within the influence of such a cir- cuit, will arrange itself at right angles to the direction of the flow, and under the same circumstances the same end of the needle will always point in the same direction. All this is the result of observation and ex- periment. 263. The earth's crust is one vast thermo-electrical circuit, and its source of heat is the sun. 264. In the diurnal motion of the earth, the different portions of its tropical regions are heated in succession by the sun during the day, and cooled by radiation during the succeeding night. The hotter portions will therefore lie to the east and the colder to the west of the sun's place. A perpetual flow of electricity is thus developed and maintained in and about the earth's crust from east to west, and gives rise to the earth's magnetic action. 265. Were the materials of the earth all equally good electrical con- ductors, and the sun always in the equinoctial, the electrical flow would be parallel to that great circle, and the compass-needle would always point directly north and south. But neither of these conditions obtains. The materials vary greatly in conducting power, and the sun's declination is ever changing. 266. The disparity of conducting power directs the electrical flow in paths of double curvature, of which the general direction is parallel to the equator, and the varying declinations of the sun are perpetually shift- ing their precise location and shape as well as changing the intensity of the flow. 267. The position of stable equilibrium, assumed by a magnetic nee- dle reduced to its axis, freely suspended from its centre of gravity, and sub- jected alone to the directive action of the earth's magnetism, is called the magnetic position of the place. 268. The intersection by a vertical plane throifgh the magnetic posi- tion with the celestial sphere, is called the magnetic meridian. 269. The angle made by the magnetic and the true meridian is called the magnetic declination, or simply declination. 270. The inclination of the magnetic position to the hrrizon is called the magnetic inclination or dip. 271. The magnetic position at the same place is continually varying Tt describes daily a conical surface, of which the place is the vertex, and (J4 SPHERICAL ASTRONOMY. daily mean position the axis, while this axis itself describes a similar sur- face once a year about an annual mean position. 272. The mean of all the declinations and of dips throughout any one day are the declination and dip for that day, and are called th< diurnal declination and dip. The mean of all the diurnal declination* and dips for the different days throughout any given year, are the decli- nation and dip for that year, and are called the annual declination and dip. 273. The daily and annual fluctuations here referred to are called periodic changes. The annual de.clination and dip also change, and these changes, which are found to take place in the same direction for a great many years, are called secular changes. 274. The magnetic declination and dip vary, in general, with the locality. The line connecting those places where the declination is zero, is called the line of no declination ; and the line through the placer, where the dip is zero, is called the magnetic equator. Fig. 5T. 275. According to the Magnetic Atlas of Hansteen, constructed for 1787, the line of no declination is found on the parallel of 60 north, a little to the west of Hudson's Bay; it proceeds in a southeasterly direc- tion, through British America, the northwestern lakes, the United States, and enters the Atlantic Ocean near Chesapeake Bay, passes near the An- tilles and Cape St. Roque, and continues on through the southern Atlantic till it cuts the meridian of Greenwich in south latitude 05. It reappears in latitude 60 south, below New Holland, crosses that island through its centre, runs up through the Indian Archipelago with a double sinuosity, and crosses the equator three times first to the east of Borneo, then be- tween Sumatra and Borneo, and again south of Ceylon, from which it passes to the east through the Yellow Sea. It then stretches across the TLKRESTRIAL MAGNETISM. 05 coast of China, making a semicircular sweep to the west till it reaches the parallel of 71 north, when it descends again to the south, and re- turns northward with a great semicircular bend, which terminates in the White Sea. On the magnetic chart this line is accompanied through all its windings- l>y other lines upon which the declination is 5, 10, 15, &c. ; the latter becoming more irregular as they recede from the line of no declination. The use of these lines is to point out to navigators sailing by compass, the- bearing of the true meridian from the magnetic. 276. On the east of the American and west of the Asiatic branch of the line of no declination, the declination is west, while to the west of the American and east of the Asiatic branch the declination is east. 277. The magnetic equator cuts the terrestrial equator, according to Ilansteen, in four, and to Morlet in two points, called nodes, one of which is in the centre of Africa. 278. Beginning at the African node the magnetic equator advances rapidly to the north, and quits Africa a little south of Cape Guardafui, and attains its greatest north latitude, 12, in 62 of east longitude from Green- wich. Between this meridian and 174 east, the magnetic is constantly to the north of the terrestrial equator. It cuts the Indian peninsula a little to the north of Cape Comorin, traverses the Gulf of Bengal, making a slight advance to the terrestrial equator, from which it is only 8 distant a its entrance into the Gulf of Siam. It here turns again a little to the north, almost touches the north point of Borneo, traverses the straits be- tween the Philippines and the isle of Mindanao, and on the meridian of Naigion it again reaches the north latitude of 9. From this point it traverses the archipelago of the Caroline Islands, and descends rapidly to the terrestrial equator, which it cuts, according to Morlet in 174, and according to Hansteen in 187 east longitude. Its next point of contact with the equator is in west longitude 120. Here, according to Morlet, it does not pass into the northern hemisphere, but bends again to the south, while Hansteen makes it cross to the north, and continue there for a dis- tance of 15 of longitude, and then return southward and enter the south- ern hemisphere in longitude 108 west, or 23 from the west coast of America. Between this point and its intersection with the terrestrial equator in Africa, the magnetic equator lies wholly in the southern hemi- sphere, its greatest southern latitude being about 25. 279. The dip increases as the needle recedes on either side from the magnetic equator, the end of the needle which was uppermost in the northern being lowermost in the southern hemisphere. 5 C6 SPHERICAL ASTRONOMY. 280. The points at which the magnetic needle is vertical are called the magnetic poles. Of these there are four, two in each hemisphere, their positions being indicated on the magnetic charts. 281. On the magnetic charts, the magnetic equator is accom pauied by curves of equal jp as in the case of the lines of equal decli nation. 282. The line of no declination and the nodes of the magnetic equa- tor are found to have a slow westerly motion, thus causing the differ- ent lines of equal declination and dip to pass successively through the same place, and illustrating the utter worthlessness of all maps constructed from compass bearings unless the diurnal declinations of the needle are carefully ascertained and recorded thereon. 283. The intensity of the earth's magnetic action increases with the proximity of the electrical paths to the needle and with the difference o( temperature in their different parts ; and from changes in these, produced by the varying zenith distance of the sun during the day, and of his me- ridian zenith distance throughout the year, arise the daily and annual mutations of declination and dip ; while to changes of the earth's crust, produced by geological causes, and increased cultivation of the soil from the spread of civilization, are to be attributed the secular variations of the same elements. TIDES. 284. Those periodical elevations and de- pressions of the ocean by which its waters are made to flow back and forth through the estuaries that indent our coasts, are called Tides. 285. Perpetual change in the weight of the waters of the ocean, due to the attraction of the sun and noon upon the earth, and the diurnal rotation of the latter about its axis, cause and mairtvn the tides. 28G. Let AGED be a great circle of the earth, in a plane through the sun's centre at S. Draw S E through the earth's centre at E, and CD through the same point, and at right angles to S E. Assume any unit of mass as that at #; join G and S, and make TIDES. (J7 d = S E = distance of sun from the earth ; p = E G ~ radius of the earth ; z = S G = distance of G from sun ;

+ _ P .sin 2

lhat of the sun will be 2A:pra355000x88 a ~(400) 3 1 3 * and dividing the last by the first, we have 355000 X 88 (400) 3 = 0.488 ; so that the effect of the moon is more than double that of the sun. 293. The lunar day exceeds the solar on an average about 50 min- utes ; the lunar tide must therefore move slower than the solar by about 1 2.5 in 24 solar hours ; and hence they must sometimes conspire and sometimes oppose one another. The former occurs when the angular dis- tance of the sun from the moon, as seen from the earth, is or 180, and (he latter when this distance is 90. This alternate reinforcement and partial destruction of the lunai by the solar wave, produce what are called spring and neap tides ; the former being their sum, the latter their difference. 294. The sun and moon, by virtue of the ellipticities of the terres- trial and lunar orbits, are alternately nearer to and further from the earth than their mean distances. TIDES. 71 If the mean distances of the sun and moon be substituted in Eq. (80), the corresponding ellipticities of the solar and lunar spheroids will be found to be 2 and 5 feet respectively ; so that the average spring tide will be to the average neap, as 5 + 2 to 5 2, or as 7 to 3." Substituting the greatest and least distance of the sun in the same equation, the resulting tides are called respectively apoyean and pe^igean tides ; and representing the ellipticity of the solar spheroid at the mean distance by 20, the corresponding ellipticities become 19 and 21. In like manner the ellipticities of the lunar spheroid will be found to vary be- tween the'limits 43 and 59. Hence, the highest spring tide will be to the lowest neap, as 59 + 21 is to 43 21, or as 10 to 2,8. 295. The sun and moon act to form the apexes of their respective tide-waves at different places, depending upon their angular distances apart This gives rise to a resultant wave, whose apex is at some inter- mediate place, and the actual tide day, or interval between the occurrences of two consecutive maxima of the resultant w#ve at the same place, will vary as the component waves approach to or recede from one another. This variation from uniformity in the length of the tide day is called the priming or lagging of the tides the former indicating an acceleration and the latter a retardation of the recurrence of high-water at the same place. The priming and lagging are particularly noticeable about the time the angular distance between the moon and sun is or 180, that is, as we shall presently see, about new or full noon. 296. The effort of the attracting body being to form the nearest ver- tex of its aqueous spheroid immediately under it, the summit of the lunar and solar tide-waves follow the course of the moon and sun to the north and south of the equator, and this gives rise to a monthly and annual variation in the heights of the pnncipal tides at a given place. 297. But of all causes of difference in the heights of tides, local situation is the most influential. In some places, the tide-wave rushing up narrow channels becomes so compressed laterally as to be elevated to extra- )rdinary heights. At Annapolis, in the Bay of Fundy, it is said to rise 120 feet. 298. Were the waters of the ocean free from obstmctions due to viscosity, friction, narrowness of channels leading to different ports, and the like, the time of high-water at a given place, would depend only upon the relative positions of the sun and moon, and their meridian passages. But all these causes tend to vary this time, and to postpone it unequally at different ports. This deviation of the time of actual from that of theoret- ical high-water at any ph;ce, is called the establishment of the port, and is 72 SPHERICAL ASTRONOMY. an element of the highest maritime importance. When ascertained from observation, it enables the manner to know by simply noticing the places of the sun and moon with reference to the meridian, when ne may safely attempt the entrance of a port obstructed by shoals. 299. In bays, rivers, and sounds, where tides arise from an actual flow of water, the time of " Slack water" or stagnation, must not be con- founded with that of high and low water. They may, indeed, coincide, but not of course. A river current, for instance, and another from the sea, ..., neutralize each other's flow, while both conspire to elevate the water surface; so, also, an ebbing current may continue its onward course after the more advanced part of a returning flood has put its surface on the rise by checking its velocity. The same of two currents meeting in a sound. 300. Starting from A as an origin (Fig. 58), and proceeding in the direction of A C B D A, we find the value of T, Eq. (81), negative in the 1st and 3d quadrants, and positive in the 2d and 4th ; so that th* tangential components of the solar and lunar attractions conspire with the normal to increase the height of the gpeat tide-waves by impressing upon the water a motion of translation towards their apexes. But before the inertia of the water will permit the latter to acquire much velocity, the rotary motion of the earth reverses the direction of the impelling forces, and the final effect due to this cause is, in consequence, but small. TWILIGHT. 301. The curve along which a conical surface, tangent to the sun and earth, is in contact with the latter body, is called the circle of illumination. It divides the dark from the enlightened portion of the earth's surface, and is ever shifting its place by the diurnal motion. 302. The base of the earth's shadow, into which a spectator enters at sunset, and from which he emerges at sunrise, is inclosed by an atmospheric wall-like ring, illuminated by the direct light from the sun, immediately exterior to that which just grazes the earth's surface. The light is reflected from the particles of this ring into the shadow, and gives to the air about its boundary a secondary and partial illumination called Twilight. A co- nical surface through the summit of this ring, and tangent to the earth, determines, by its contact with the latter, a limit within which the twilight cannot sensibly enter, and twilight will only continue while the spectatoi is carried by the earth's diurnal motion across the zone of which this line is the inner, and the circle of illumination the exterior bounda-v. The TWILIGHT 73 belt of the earth's surface over which twilight is visible, is called the cre- puscular zone. Thus, let E O f E' be a section Fi ?- 60 - of the earth's surface on the opposite side from the sun ; TAA' T' of the atmosphere by the same plane, the height of the air being exaggerated to avoid confusing the figure; and S A and S' A' two solar rays tan- gent to the earth's surface. The particles of air in EA T and E'A'T will be illuminated, while those in fbe space EAA'E' will be in the shadow. The section will cut from the tangent cone the elements A V and A' V, which touch the earth at and 0', respectively, and being revolved about the line connecting the centres of the earth and sun, the part EA T will generate the lumin- ous atmospheric inclosure and the points E and 0, the circle of illumina- tion and interior boundary of the crepuscular zone, respectively. 303. To a spectator within the crepuscular zone a portion only of the illuminating ring will be visible, and will appear as a bright elliptical seg- ment, with its chord in the horizon, its vertex in the vertical circle through the sun, and its outline almost lost in the gradual decay of light produced by the diffusive action of the air and the progressive thinning and conse- quent diminution in the number of reflecting particles towards the summit of the luminous ring. 304. When the spectator is carried obliquely through the crepuscular zone without crossing its smaller base, twilight will last all night. 305. Resuming Eq. (74), that is cos z = sin I sin d -f cos I cos d cos P ; substituting the latitude of the place for , the declination of the sun for rf, and the value of P, obtained by converting the observed time from noon to the end of twilight in the evening, or from the beginning of twilight in the morning till noon, into degrees, the average value of a number of de- terminations for z will be found to be about 108; so that at the end of evening or beginning of morning twilight the sun is 18 below the hor'zoiL 306. From the above equation we find 74 SPHERICAL ASTRONOMY. COS 2 COS I . COS d . COS P sn = sin d The angle P S Z, made by the hour circle P S and vertical circle Z S, is called the variation or the parallactic angle. Denote this by , then from the triangle Z P S, will ( 1 ) . . . sin I =. sin d cos z -f- cos d sin g cos Equating the second members of this and the equation above, we have (2) .... cos I . cos P = cos z . cos d sin z sin d . cos ; and if the sun be in the horizon, then will z = 90, P = P', and ( = ?, and (3) .... cos I . cos P' =. sin d . cos '. Also, from the same triangle, * (4) .... cos I. sin P = sin z .sin ; and when the sun is in the horizon, (5) . . . . cos I . sin P' = sin f . Multiply (2) by (3), also (4) by (5), and add the products, there will result, cos I . cos (P P 1 ) = cos z cos d sin d cos f -f sin z cos ( f ) cos d sin a cos cos f . From (1), we have sin / sin d . cos z cos = cos a . sin and for the sun in the horizon COS P i= sin / (82) (83) TWILIGHT. which substituted above, give J/RNM. cos 2 / . cos (P P') = sin z . cos ( ') sin 8 / ; whence, because cos (P - P') = 1 - 2 sin 2 \ (P - P'), we have 1 sin z . cos (B ') ^ sin 2 J (P - P') = 2 cos 2 passing to the arc and making _P P' ~15~' we have which will give the time required for the sun, or other heavenly body, to pass from the horizon to a zenith distance z, or, conversely, from a zenith distance z to the horizon. Making z = 90 -f 18 = 108, Eq. (84) becomes which will give the duration of twilight for any latitude and season of the year ; and for this purpose, the values of and ' must be found from Eqs. (82) and (83), after making, in the former, z = 90 + 18. The value of /, in Eq. (85), becomes a minimum when = ', and for the duration of the shortest twilight, we have, after replacing 1 cos 18 by its equal 2 sin 2 9, t . sin" 1 (sin 9 . sec /) (86) lo Equating the second members of Eqs. (82) and (83) sin d - - tan 9 . sin J (87) In a given latitude, Eq. (86) will make known the shortest twilight, and Eq. (87) the season at which it will occur. * Ann Arbor Astronomical Notices, N .'I. 76 SPHERICAL ASTRONOMY. 308. The sign of the second member of Eq. (87) shows that at the time of shortest twilight the spectator and the sun will be on opposite sides of the plane of the equinoctial. 309. The depression of the lowest point Q' of the equinoctial below the ho- rizon HH', is 90 I ; and of the low- est point S of the sun's diurnal path, when his declination is of the same name as the spectator's latitude, 90 (I + d) ; and when 90- = 18, the end of the evening will be the beginning of morning twilight, and the nocturnal path of the spectator will be tangent to the inner boundary of the crepuscular zone. THE SUN. 310. The Sun, as before stated, is the central body of the solar sys- tem, and from this circumstance gives to the latter its name. It occupies one of the foci of all the elliptical orbits of the planets, and, of course, that of the earth. 311. Distance and Dimensions of the Sun. Its horizontal parallax denoted by P, and apparent semi-diameter denoted by s, vary inversely as the earth's radius vector. For the mean radius it is found, 113-6, P = 8".6, ands = 16' 01".5; which in Eqs. (28) and (29) give w 206264".8 = P-p = P- 16' 01".5 _ "8^6" ~ P "7T- = 23984 ' P 961".5 8".6 = 111.5 p (88) . . (89) From Eq. (88) it appears that the mean distance of the earth from the sun is 23984 times the earth's equatorial radius ; and from Eq. (89) thai the sun's diameter is 111.5 times that of the earth. The volumes of these bodies are as the cubes of their diameters, and hence the volume of the sun is 1384472 times that of the earth. THE SUN. 77 g 312. If the equatorial radius p be replaced in Eqs. (88) and (89) by its value in miles, 98, we find r n = 95,043,800 miles, 2d= 882,000 " ; that is to say, the mean distance of the earth from the sun is, in round numbers, about 95 millions of miles, and the diameter of the sun is 882 thousand miles. The mean distance of the earth from the sun is assumed as the unit of linear dimensions in all celestial measurements. 313. Mass of Sun. In Analytical Mechanics, 201, we find the equation (89)' in which T denotes the periodic time of a body revolving about a centre of attraction, a the mean distance of the body from the centre, if the ratio of the circumference to the diameter, and k the attraction on a unit of mass at the unit's distance. Let k become fjo in the case of the sun's action on the earth ; then will T become the sidereal year, and a the semi-transverse axis of the earth's orbit, and -^ ........ (90) and for the action of the earth upon the moon in which jx' denotes the^ attraction on the unit of mass at the unit's distance exerted by the earth. Now the attractions exerted by two bodies on the same mass at the same distance, are directly proportional to their masses respectively ; and denoting the mass of the sun by M, and that of the earth by M' we have -t-r ., (92) M 1 ~~ ' ..... (93) THE SUN. 79 and substituting the values just found, W ^ = 28,5. That is, a body weighing one pound at the* equator of the earth \*culd weigh 28,5. pounds at that of the sun; and acquire, therefore, during each second of its fall a velocity of 916,44 feet. 316. Sun's Rotation and Axis. Through the telescope Fi &- 6S - the sun's surface often exhibits dark spots which slowly change their places and figure. They cross the solar disk from east to west, and thus reveal a rotary motion of the sun itself from west to east about an axis. 317. To find the time of rotation and the position of the axis, it will be necessary first to find the heliocentric longitudes and latitudes of the same spot at different times. To do this, let S be the sun's centre, E that of the earth, P the spot, and N its projection upon the plane of the ecliptic Maks I heliocentric longitude of the earth ; x= " " " spot; y = P S JV = heliocentric latitude of spot ; <3 = P EN = geocentric latitude of spot ; e = SEN = difference of geocentric longitude of the sun and the spot, J = sun's apparent semi-diameter. Then SP sin y = P N = EP sin /3 = SE sin /3, because the difference between EP and S E is insignificant in comparison with either ; whence SE _ sin (3 Again SP . cos y : EP . cos ft : : SN : NE, : : sin e : sin (/ whence sin e . cos (3 EP sm x = cos y sin e . cos j8 ^ sin 4 . cos y ' and replacing cos y by its value, sin e . cos 8 sm (/ x) = - ____ ; 2 - 9 ' 80 SPHERICAL ASTRONOMY. or for logarithmic computation, sin e . cos (3 sin (I x) = . -V/sin (J + ) . sin (^ - /3) 318. Position of the Suns equator, and the time of the Sun's rota- tion. Let E be the pole of the ecliptic, P that of the sun's equator ; A. A', and A" the heliocentric places of the same spot observed at three different times ; and let E A, E A \ E A" , PA, PA', PA" he the arcs of great circles. The first three are known from Eq. (94), being the helio- centric colatitudes of the spot ; as also the angles A E A' , AEA", and A' E A" from Eq. (95), being the differences of the he- liocentric longitudes all deduced from ge- osurface observations of the spot's right ascension and declination, 152. All the sides and angles of the triangles AE A', AEA", and A'EA" may be found, two sides and the included angle in each being given ; hence the sides A A', A' A", and A" A, and the angles A, A', and A', in the triangle A A' A", are known. Now P being the pole of the sun's equator, parallel to which the spot revolves, PA = PA'= PA"-, Make 2S = A + A' + A" = 2P AR + 2PA' A -f 2PA' A" = 2PAR + 2A f : whence PAR = S - A', and PAR becomes known. If PR be perpendicular to AA" , then in the right-angled triangle APR, the angle at A and the side AH being known, the side PA is computed j and, finally, in the triangle APE, the sides AP and AE, and the angle EA P = EAA" -PAA" being known, P E is computed. 319. The arc EP is the heliocentric colatitude of the pole of the sun's equator, and the angle AE P, added to the heliocentric longitude of the spot at A, gives its heliocentric longitude. The position of the sun's equator becomes, therefore, known. The heliocentric latitude and longi- tude of its north pole at the beginning of the present century were, respec- tively, 82 30' and 350 21'. THE SUN. 81 Fig. 65. From the triangle APR the angle AP R becomes known, the double of which is AP A". Then, denoting by T the time of one rotation, and by t the interval between the observations on the spot at A and A", we have A PA" : t :: 360 : T] whence Tis known to be about 25.325 days, making the angular velocity of the sun around its axis about one twenty-fifth that of the earth. From this motion it is concluded that the sun is flattened at its poles. 320. Physical constitution of Sun. 'ihe study of the solar spots has led to inter- esting conclusions in regard to the physical constitution of the sun itself. The spots are transient in character, variable in size, shape and number, and confined to two compara- tively narrow zones parallel to, and at no great distance from the sun's equator. They appear perfectly black, and surrounded by a border less dark, called a penumbra. The black part and penumbra are distinctly de- fined in outline, and do not fade the one into the other. F1 - 66 - Sometimes this penumbra presents two or more shades, and in this case also there is no gradation, but well-marked out- line, indicating a total absence of blending. As the spots move towards the edge ri &- 67 - of the sun, the penumbra on the inner :: ^ " -- side gradually contracts, and with the black spot disappears before reaching the boundary of the disk ; the penum- bra on the outer side expands, and is the last visible remnant of the spot as it passes behind the sun. At its reappearance on the opposite edge of the sun, the spot exhibits similar phenomena the penumbra first appears, then the black portion on its in- ner side, the contraction of the penumbra in width, and its extension* around the black till the latter is entirely surrounded. This is precisely the appearance that would be presented by a deep pit or excavation with a dark or non-luminous bottom. The rotation of the sun would bring the slanting surface leading from the inner edge of its mouth more and more in the direction of the spectator till it would be lost in the foreshortening, the inner edge would presently mask the bottom, and the surface of the opposite side would be turned so nearly perpendicu- 6 82 SPHERICAL ASTRONOMY. larly to the line of sight as to appear broadest just before passing behind, at disappearance, or at reappearance, to the front of the sun. 321. The spots gradually expand or contract, change their figure, vanish, and break out again at new places where none were before. When Fig. 68. disappearing, the central black part contracts to a point and vanishes be fore the penumbra ; and a single spot is sometimes seen to break up ink two or more smaller ones. 322. A circle of which the diameter is one second is the smallest vis- ible area. A single second at the earth is subtended at the sun by a dis tance of 461 miles, and the area of the least visible circle on the sun's surface is, therefore, 167,000 square miles. A spot whose diameter was 45,000 miles has been known to close up and disappear in course of six weeks, thus causing the edges to approach one another at the rate of 1000 miles a day. Many, spots distinctly visible have been observed to vanish in a few hours, indicating a degree of mobility inconsistent with the idea of solids and liquids. 323. Light proceeding very obliquely from the surfaces of incandes- cent solids and liquids is always polarized, whereas that from gases under the same circumstances is not. The light from the edge of the solar disk THE SUN. 83 leaves the surface of the sun iu a direction nearly coincident with the surface itself, and yet when examined by the usual tests exhibits no signs of polarization. 324, The luminous part of the sun is not uniformly bright, but pre- sents a mottled appearance, and immediately about the spots are often seen well-defined and branching streaks, called facules, brighter than other parts of the surface ; among these, spots often make their appear- ance. They are best seen near the border of the disk. 325. The brightness of the solar disk sensibly diminishes towards the borders ; and this fact has given rise to the supposition that the sun is surrounded by an atmosphere not perfectly transparent, and of great extent above the luminous envelope. The loss of light towards the bor- ders would result from the greater absorption of the luminiferotis waves in consequence of traversing a greater thickness of the atmosphere in that direction. 326. The moon, of which an account will be given presently, is known to be a non-luminous, opaque, spherical mass, and so near the earth as to give to it an apparent diameter about equal to that of the sun. This little body often interposes itself so as completely to conceal the sun from view, producing what is called a solar eclipse. At the in- stant of greatest solar obscuration that is, when the rnoon completely covers the sun red protuberances resembling flames of fire are seen to issue apparently from the edge of the moon, but in fact from that of the sun, revealing the existence of intense commotion and physical changes about the surface of the latter body. 327. From all which it is inferred that the sun is an opaque solid, cov- ered by a gaseous envelope of well-defined boundary and intense luminosity, the whole being surrounded by a non-luminous atmosphere of vast extent. No explanation free from objection has, thus far, been given for the solar spots. Some have supposed them to arise from scoria or flakes of incombustible matter floating upon the sun's surface; while others, with perhaps greater reason, have attributed them to temporary openings in the photosphere that envelops the sun, exposing to view detached por- tions of his solid crust, which appear black from contrast. But it must not be inferred from this that the solid portion of the sun is regarded as non-lnmiuous. Were he stripped of his gaseous coating, he would no doubt shine with diminished but yet intense brilliancy. A piece of quicklime, in a state of most active combustion under the action of a compound blowpipe, is, when projected upon the bright part of the sun, as dark as the darkest part of the spots* During the interposition of the lunar sc:een between the sun and ; 84 SPHERICAL ASTRONOMY. spectator on the earth, the surrounding landscape takes on the obscure illumination produced by a closing evening twilight, and the temperature is always sensibly depressed, thus corroborating the suggestions of other phenomena, that the sun is the great source of light and heat to the earth. But light and heat are the results of molecular agitation. What, then, is the cause of that perpetual molecular vibration essential to the self-luminosity of the sun ? The solar system is believed to have resulted from the subsidence of a vast nebula; the planets and satellites are de- tached fragments left behind in the progress of the general mass towards the centre ; the sun itself is the central accumulation. This nebula must have extended originally far beyond the orbit of Neptune, the ex- tenor planet now known. The distance of this planet from the sun is more than thirty times that of the earth. The condensation has taken place under the action of weight impressed upon the elements by their reciprocal attractions for one another. The living force with which so much matter would reach the terminus of a fall necessary to transfer it to its present abode, could not fail to impress upon the condensed mass the most intense molecular agitation. This agitation, or molecular liv- ing force, can only be lost through the agency of the surrounding me- dium which diffuses it through space; and the loss in a given time is determined by the density of the medium, being less as the density is less. The medium which pervades the planetary space is so attenuated ? V& and denoting the mean mati/ = N'R" = distance of perihelion in right ascension from the asc. node. vt = B R" = heliocentric right ascension of the perihelion. 88 Then SPHERICAL ASTRONOMY. tan X = (113) tan P"SB = tan P'QP" - cot P' C A = -; x and in the triangle A P' C, the side AC being 90, cot d = cosX- - (114) Again, in the triangle P'P"W, right-angled at P'", sin v\ = cot 8 - cot i (115) e = X ) (116) tan 9 = sec t* tan (X s) (117) In the triangle ITR'R", right- angled at 72", tanX'= cos t.tan ( and this in Eq. (29) gives rf-P-^7 (122) in which s is the planet's apparent semi-diameter measured with the mi- crometer, d its real semi-diameter, and p the earth's equatorial radius ; whence the diameter, surface, and volume of the planet become known. 354. MERCURY and VENUS are called inferior planets, being lower or nearer to the sun than the earth; the others are called superior planets, because they are higher or more distant from the sun than the earth. 355. When the geocentric longitude of a body is the same as that of the sun, the body is said to be in conjunction; when its longitude differs by 180, in opposition. The superior planets may be in opposition, but the inferior planets never. 356. A body in conjunction or opposition is also said to be in syzygy. PLANETS. Fig. Ti. . 357. When an inferior planet is in perigean syzygy, it is said to be in inferior conjunction ; when in apogean syzygy, in superior con- junction. 358. Synodic revolution. The interval of time between two consecutive returns of a planet to apogean or perigean syzygy is called its synodic revolution. Denote by m the heliocentric mean daily motion of the earth in longitude ; by TI, that of any planet ; and by T, the length of its synodic revolution ; then will m ~ n be the relative motion in longitude of the earth and planet, and 359. Geocentric Motion in Longitude. The angle at the earth, subtended by a body's linear distance from the sun, is called the body's elongation ; the projection of a body's centre on the plane of the ecliptic, is called the reduced place ; and the projection of its radius vector, is called the curtate distance. Thus, let S be the sun, P a planet, E the earth, and P N a perpendicular from the planet to the plane of the ecliptic, intersecting the latter in N; then will SEP be the elongation, N the reduced place, and S N the curtate distance of the planet. 360. Draw S V and E V to the vernal equinox ; they will be sensibly parallel. Also drawJVA 7 / and E E t perpendicular to E V 3 and S F", and make a = S ~N = mean curtate distance ; p = EN = earth's distance from the reduced place ; / = VS JV= planet's heliocentric longitude ; n = hourly change in the same ; L = V S E = earth's heliocentric longitude ; X = V EN= planet's geocentric longitude; m hourly change in the same. 92 SPHERICAL ASTRONOMY. Then, the mean distance of the earth from the sun being unity, will. Appendix X., m = P z [a 2 + a2 _ (a + a?) . cos (L 1)] . n . . . (124) m which cos X p a cos I cos L and which will make known the rate and direction of the body's motion in geocentric longitude. 361. Direct and Retrograde Motion ; Stations. When the planet is in apogean syzygy, then will L I = 180, cos (L 1) = 1 ; and, Eq. (124), m = P 2 .a.(a + I) (1 +a*).n ..... (125) and m will always be positive ; that is, the geocentric motion of the planet will be direct. 362. When the planet is in perigean syzygy, then will L I = ; cos (L 1) = 1 ; and, Eq. (124), a).n ...... (126) and m will always be negative, whether a be greater or less than unity ; that is, the geocentric motion of the planet will be retrograde. 363. In changing from direct to retrograde, and the converse, the body must appear stationary. This will make m = 0, and, Eq. (124), coB(Z-J) = - - ^ = _- - I -- ...... (127) 1 + 2 a 2 + a~ 2 1 a quantity which is always less than unity, whether a be greater or less than unity ; that is, all the planets must sometimes appear stationary. The condition expressed by Eq. (127), may always be satisfied for two val- ues of L I. The two places of a body, in which it appears stationary, a e called stations. 364. Let the value of L I for one of the stations be

If P and P / be whole numbers, and the second member be reduced to its simplest terms, the numerator will be the interval in sidereal years between, the consecutive transits at the same node, and this interval will be constant. But if P and P / be not whole numbers, then will the numerators of the- approximating fractions of the continued fraction, which give the values of the second member within the transit limits, be the variable intervals, in sidereal years, between the transits at the same node. 375. Masses and Densities of the Planets. The masses of such of the planets as have satellites may easily be found by the process of 313, as Boon as the periodic time of the planet and that of its satellite are deter- mined by observation. But for such as have no satellites, recourse is had to a different process, which can be here indicated only in outline. A 98 SPHERICAL ASTRONOMY. planet undisturbed by the action of the others, would describe accurately its elliptical orbit about the common centre of inertia due to its own mass and that of the sun ; and from the elliptical elements already described, its future places are, as we shall see, predicted with the greatest precision. Tha difference between these places and those actually observed, give the effects of the disturbing action of the other planets. To compute these effects, what are called perturbating functions are constructed upon the principles of mechanics. The masses of the perturbating or disturbing bodies enter these functions ; and from the observed amount of perturb- ations the value of the masses are computed. An. Mec., 203, 376. The masses and volumes being known, the densities result from the process of 314. 377. Rotary motions. All the planets whose surfaces exhibit through the telescope distinct marks, are found to have a rotary motion in the same direction as those of the sun and earth, viz., from west to east. 378. Planetary Atmosphere. The existence of an atmosphere about a planet is indicated by the apparent displacement it occasions in the geo- centric place of a star by refracting its light, when, by the motion of the earth and planet, the latter comes near the line of the star and observer. The atmosphere about a planet is in fact a vast spherical lens, of which the central part is deprived of its transparency by the opaque materials of the planet, but of which the outer portion is free from obstruction and acts upon the light which passes through it with an energy due to its refractive power and density. The height of the atmosphere is inferred from the greater or less angular distance between the star and planet when the displacement begins ; and the density, which must be regulated by the same laws that govern the equilibrium of heavy elastic fluids upon the earth, from the amount of dis- placement. 379. In detailing the physical peculiarities of the planets, their mean distances MTU! times of sidereal revolutions, although contained in the sy- noptical table of elements, will be repeated ; and in all cases in which di- mensions or measures are given, they must be understood as expressed in the corresponding elements of the earth as unity. Thus, if it be the mean distance, density, volume, solar heat and light, sidereal day, rtional to the true magnitudes, the ra- dius of the earth viewed at the distance of the sun, in other words, the sun's hori- zontal parallax, is equal to the angular distance between the chords divided by 5.22. 391. The relative geocentric motion of the sun and planet into the obserred durations of the transit at the two stations will give the chords m n and p q. The chords being known, as also the apparent 102 SPHERICAL ASTRONOMY. semi-diameters Sq and S n, the distances SatmASb become known, and therefore their difference a 6. 392. The general result of all the observations made on the transit of 17G9 gives 8''.5776 for the sun's horizontal parallax. The next two transits of Venus will occur on Dec. 8th, 1874, and Dec. 6th, 1882. MARS. 393. Mars is the first of the superior planets. His mean distance is 1.5237; sidereal year, 1.8807; true diameter, 0.517; volume 0.1380; density, 0.95 ; equatorial gravitation, 0.493 ; solar heat and light, 0.43 ; sidereal day, 1.02694 ; oblateness, about 19 ; and the inclination of his axis to that of his ecliptic 30 18' 10".8. 394. He has a dense atmosphere of moderate height. His surface (Plate II., Fig. 2) exhibits through the telescope outlines of what are deemed to be continents and seas, the former being distinguished by a ruddy color, which is characteristic of this planet, and indicates an ochry tinge in the soil, contrasted with which the seas appear of a greenish hue. These markings are not always equally distinct ; and the variation is attributed to the formation of clouds and mists in the planet's atmosphere. Brilliant white spots sometimes appear at that pole which is just emerging I'rom the long night of its polar winter, and are attributed to extensive snow-fields that push their borders to an average distance of some six de- grees from either pole. PLANETOIDS. 395. Next to Mars come the class of small planets, which, on account of their comparatively diminutive size, are called planetoids. Little is known of them beyond their orbit elements, but they are interesting on account of their history and the speculations connected with their discov- ery, which began with the present century. , 396. If the mean distance of Mercury be taken from the mean dis- tances of the other planets, the remainders will form a series of numbers doubling upon each other in proceeding outward from the sun. To this law there was a remarkable exception in the distance between the orbits of Mercury and Jupiter as compared with that between Mercury and Mars, the former being so large as to require the interpolation ot another body between Mars and Jupiter. 397. Although th * law is strictly empirical and wholly inexplicable PI ate HI. TOFKONT -PAGE PLANETOIDS. 103 a priori upon any known physical hypothesis, yet the coincidence was so remarkable as to induce the prediction that by proper search a planet would be found in the 'interpolated place. 398. This body was only to be recognized by its proper motion. 1\ detect this, an examination of the telescopic stars of the Zodiac was com- menced, their places were carefully mapped, and on the first day of the present century, the prediction was verified by the addition of Ceres to the system. Her mean distance is 2.76692, and the hiatus was filled. 399. But the discovery of Ceres was soon followed by that of Pallas, at the mean distance of 2.7728 nearly the same as that of Ceres and the law was again broken. 400. The points in which the paths of the new planets are intersected, 3n either side of the sun, by the line common to the planes of both orbits, are not very far apart, and it was suggested that Ceres and Pallas were but fragments of a larger planet that once -revolved at an average distance, and which had been broken to pieces by some disruptive force. But where were the other fragments ? 401. A number of bodies projected in different directions from a com- mon point, would each describe about the sun an hyperbola, a parabola, or an ellipse, depending upon the relations between the velocity of projection and the intensity of the sun's attraction upon the unit of mass, and in the case of elliptical orbits, the bodies would, abating the effects of the pertur- bating action of the other planets, return at fixed intervals to the place of departure. 402. The opposite points of the heavens, in which the orbits of Ceres and Pallas approached most nearly each other, were therefore regarded as the common haunts of the suspected fragments, and the places especially to be watched, to delect their existence. A constant scrutiny of these points, and diligent revision of the maps of the zodiac, have resulted in the discovery, to the present time, of 91 of these little bodies. 403. The mean distances of the planetoids vary about from 2.2 to 3.6, and periodic times about from 3.3 to 6.9. Their small size makes it diffi- cult to determine their true dimensions, the diametor of the saYne individ- ual, as given by the best authorities, varying from 0.02 to 0.20. They exhibit considerable variety of color ; some have shown signs of possessing atmospheres, and those who regard them as debris of a single body, find evidence of an angular or fraginental figure in sudden changes of illumina- tion, which have been observed, and which are attributed to the shifting of their bounding planes by a diurnal or axial rotation. 104 SPHERICAL ASTRONOMY. JUPITER. 404. Jupiter is the largest, and except Venus, which he sometimes surpasses in this respect, the brightest of the planets. His mean distance is 5.202 ; sidereal year, 11.86; diameter, 11,2 ; volume, 1280.9 ; mass, 331.57 ; density, 0.24 but little greater than that of water; equatorial gravitation, 2.716; solar heat and light, 0.037; sidereal day, 0.41376; oblateness, 20 ; inclination of axis to that of his ecliptic, 3 5' 30". 405. The disk of Jupiter is always crossed, in a direction parallel to his equator, by dark bands or belts, presenting the appearance indicated in Plate III., fii>'. 3, which was taken by Sir John H^rscliel. These belts are not always the same, but vary in breadth and situation, though never in direction. They have sometimes been seen broken up and distributed over the whole face of the planet. From their parallelism to Jupiter's equator, their occasional variation and the .appearance of spots upon them, it is in- ferred that they exisj in the planet's atmosphere, and are composed of extensive tracts of clouds, formed by his trade-winds, which, from the great size of Jupiter, and the rapidity of his axial rotation, are much more de- cided and regular than those of the earth. 406. The great oblateness of this planet is due to the shortness of his sidereal day, and its amount agrees with that assigned by theory to give him a figure of fluid equilibrium. 407. From the small inclination of his axis to that of his ecliptic, there can be but little variation in the length of his days and nights, each of which is less than five of our hours ; and changes of seasons must be almost, if not quite unknown to his inhabitants. 408. Jupiter is attended in his circuit about the sun by four satellites or moons, which revolve about him from west to east, and present a min- iature system analogous to that of which Jupiter himself is but a single in- dividual, thus affording a most striking illustration of the effects of gravi- tation and of distance in grouping, as well as shaping the courses of the heavenly bodies. These satellites will be noticed under the head of Sec- ondary Planets. SATURN. 409. Saturn is the next in order of size as he is of distance to Ju- piter. His mean distance is 9.538850 ; sidereal year, 29.46 ; true diam- eter, 9.982 ; volume, 995.00; mass, 101.068 ; density, 0.102 little more than half that of water; equatorial gravitation, 1.014; solar heat and Plate IV. TO FRONT PAGE 1O4 Plate V. TO FHONT PAGE 105 . SATURN. 105 light, 0.011 ; sidereal day, 0.43701 ; oblateuess, 25; inclination of axis to that of orbit, 26 49', and to that of our ecliptic, 28 11'. 410. Saturn is the most curious and interesting body of the system, being attended by eight satellites or moons, and surrounded (Plate IV., Fig. 4), according to some authorities by two, and others by four, broad flat and extremely thin rings, concentric with each other and with the planet. 411. The dimensions of the rings and planet, arid the intervals as given by the advocates of but two rings, are, tf miles. Exterior diameter of exterior ring .... 40.095 = 176,418 Interior " " " . . . . 35.289 = 155,272 Exterior diameter of interior ring .... 34.475 = 151,690 Interior " " " . . . . 26.668 = 117,339 Equatorial diameter of planet 17.991= 79,160 Interval between the planet and interior ring 4.339 = 19,090 Interval between the rings 0.408 = 1,791 Thickness of ring not exceeding 230 412. The evidence of recent observations with very powerful instru inents seems, however, in favor of a division of the outer ring, as just given, at a distance less than half its width from the exterior edge, and of the existence of a dusky ring still nearer the body of the planet, and composed of materials partially transparent, and possessing but feeble powers of re- flection, resembling in these particulars a shee^ of water. And there seem good reasons for believing that the rings are not precisely in the same plane. The disk of the planet is crossed by parallel belts, similar to those ot Jupiter ; these are supposed to be due to Saturn's trade-winds. From the parallelism of the belts to the plane of the rings, it is inferred that the planet's axis of rotation is perpendicular to that plane, and this is con firmed by the occasional appearance of extensive dusky spots on his sur- face, which, when carefully watched, give the time of his rotation about an axis having that direction. 413. By watching the different shades of illumination on different portions of the rings, the latter are found to complete a revolution in their own plane once in 10 h 32 m 15', thus making their sidereal day 0.43906, which exceeds that of the planet itself by 0.00205. 414. That the rings are opaque and non-luminous is shown by their throwing a shadow on the body of the planet on the side nearest the sun, and bv the other side receiving that of the planet as shown in the figure. 106 SPHERICAL ASTRONOMY Fig. 80. 415. The axes of the planet and rings preserve their directions un- changed during their orbital motion. The plane of the rings, which is inclined to that of the ecliptic under an angle of 31 19', intersects the latter plane in a line which makes with the line of the equinoxes an angle equal to 167 31', so that the nodes of the ring lie in longitudes 167 31' and 347 31'. 416. The orbital motion of the planet causes this intersection to oscil- late, as it were, parallel to itself, in the plane of the ecliptic, through a distance on either side of the sun equal to the radius vector of Saturn's orbit ; and the period of a semi-oscillation is one-half of the planet's pe- riod, or about 15 years. Within this period the plane of the ring must pass once through the sun, and from once to thrice through the earth, depend- ing upon the initial position or place of the latter when the trace of the plane on the ecliptic touches the earth's orbit at the time of nearing the sun. 417. Thus, let S be the sun, EE'E"E'" the earth's orbit, P P' an arc of Saturn's orbit projected upon the plane of the ecliptic, P E and P' E" the traces of the plane of the rings on the same, and tangent to the earth's orbit, and suppose the motion of the earth and of Saturn to take place in the direction indicated by the arrow-heads. Draw SB parallel to P E and P' E", and make r = S P = the mean distance of Saturn ; r'= SE = " " " of earth; a = P S P' = the angle at the sun subtended by PP' : then, since the angle P S B = S P E, we have Fig. 81. =7= 9^4 = 0.1082, whence a = 12 2', SATURN. 307 which divicbd by 2' 0".6, the mean motion of Saturn, gives 350.46 days, wanting only 5.8 days of a complete year ; that is to say, the earth de- scribes nearly one entire revolution in the time during which the earth's orbit is traversed by the plane of the ring. 418. The rings are invisible when their plane passes between the sun and earth, their enlightened face being then turned from the latter body ; and the interval of non-appearance will be that between any two epochs at which the plane passes the sun and earth, and of which the effect of one is to throw these bodies on opposite and the other to restore them to the same side of this plane. 419i If the initial place of the earth be at E", nearly three days in advance of B", then will the plane itself pass the sun and earth at the same time, the earth being at B' , and these bodies could not be on oppo- site sides of the plane of the rings during its present visit to the earth's orbit. If the initial position of the earth be at E', nearly three days in advance of E, it will be at E" when the plane passes the sun ; the rings will then disappear, and continue invisible till the earth meets and passes their advancing plane, which it will do somewhere in the quadrant E" B' '; they will then reappear, and continue visible for the next fifteen years. If the earth's initial place be at E'", some days in advance of B', it will meet and pass the plane in the same quadrant, the rings will disappear and continue invisible till their plane is overtaken and passed again by the earth somewhere in the quadrant E B" ', when the plane passes the sun the earth will be in the quadrant B"E", and the rings will again disap- pear, and again become visible only when their plane is recrossed by the earth in the quadrant E"B'. Thus, with this initial place, the earth will cross the plane of the rings three times in one year, and there will be two disappearances. 420. When the plane of the ring passes through the sun, the edge of the ring alone is enlightened, and can only appear as a straight line of light projecting from opposite sides of the planet in the plane of his equa- tor, and parallel to his belts. This phase of the ring has been seen, but it requires the most powerful telescopes; and from the fact of its non-ap- pearance in a telescope which would measure a line of light one-twentieth of a second in breadth, of which the subtense at Saturn's distance is 230 miles, it is inferred that the thickness of the ring cannot exceed this latter dimension. 421. When the dark side of the ring is turned to the earth, the planet appears as a bright round disk with its belts, and crossed equato- rial ly by a narrow and perfectly black line. This can only happen when 108 SPHERICAL ASTRONOMY. the planet is less than 6 1' from the node of his rings. Generally the northern side is enlightened when the heliocentric longitude of Saturn is between 172 32' and 341 30', and the southern when between 353 32 and 161 30'. The greatest opening occurs when the heliocentric longi- tude of the planet is 77 31' or 257 31'. URANUS 422. Uranus is one of the more recently discovered planets, being only recognized as a planet for the first time in 1781, though it had often been seen before and mistaken for a fixed star. Of this planet nothing can be seen but a small round uniformly illumi- nated disk without rings, belts, or discernible spots. His mean distance is 19.18239; sidereal year, 84.01; true diameter, 4.36; volume, 82.91; mass, 14.25 ; density, 0.17 ; equatorial gravitation, 0.75 ; solar heat and light, 0.003. He is attended by six satellites, which will be noticed presently. NEPTUNE. 423. Neptune is the last known planet in the order of distance, and third in size. Its discovery dates only from 1846, though its existence had been suspected from certain irregularities in the motion of Uranus, which could only be attributed to the disturbing action of some body exterior to itself. The departures of Uranus from places assigned by the combined action of the known bodies of the system, and certain assumed conditions in re- gard to position and shape of orbit, direction of motion, and mean distance, rendered highly probable by analogy, were the data from which, by the methods of physical astronomy, was wrought out in the closet in Paris, the place of a new planet whose disturbing action would account for the unexplained waywardness of Uranus. The result was sent to an observer in Berlin, and in the evening of the very day of its receipt in the latter city, Neptune was added to the known system by actual observation. It was found within 52' of the place assigned, and its discovery, in all its circumstances, must ever be regarded as one of the greatest triumphs of modern science. 424. Neptune's mean distance is 30.0367 ; periodic time, 164.6181 ; real diameter, 4.5 ; volume, 91.125; mass, 18.219 ; density, 0.208 ; equa- torial gravitation, 0.9035 ; solar heat and light, 0.0011. The apparent size of the sun as seen from the earth, bears to that as seen SECONDARY BODIES. 109 from Neptune, about the relation of an ordinary orange to a common duck- shot. 425. Neptune has at least one satellite, and certain appearances have indicated a second, and also a ring, but of these there are yet doubts. General Remark. 426. In the foregoing enumeration of the physical peculiarities of the planets, one is impressed by the great differences in their respective sup- plies of heat and light from the sun ; in the relations which the inertia of matter bears to its weight at their surfaces ; and in the nature of the ma- terials of which they are composed, as inferred from variety of mean density. The intensity of solar radiation is nearly seven times greater on Mercury than on the earth, and on Neptune 900 times less, giving a range of which the extremes have the ratio of 6300 to 1. The efficacy of weight in counteracting muscular effort and repressing animal activity on the earth, is less than half that on Jupiter, more than twice that on Mars, and probably more than twenty times that on the planetoids, making a range of which the limits are as 40 to 1. Lastly, the density of Saturn does not exceed that of common cork. Now, under the various combinations of elements so important as these, what an immense diversity must exist in the conditions of animal life, if the planets, like our earth, which teems with living beings in every corner, be inhabited ! A globe whose surface is seven times hotter than ours or 900 times colder, on which a man might by a single muscular effort spring fifty feet high, or with difficulty lift his foot from the ground ; where his veins would burst from deficiency or col- lapse from excess of atmospheric pressure, affords to our ideas an inhospi- table abode for animated beings. But we should remember that heat and cold, light and darkness, strength and weakness, weight and levity, are but relative terms ; and to the very conditions which convey to our minds only images of gloom and horror, may be adjusted an animal and intellectual existence which make them the most perfect displays of wisdom and be- neficence. SECONDARY BODIES. 427. The secondary bodies are those which revolve about the planets, and accompany them around the sun. Of these, twenty are known at the present time. One belongs to the earth, four to Jupiter, eight to Saturn. six to Uranus, and one to Neptune. They are commonly called satellites, and sometimes moons, but this latter appellation is more particularly ap- plied to the earth's secondary. 110 SPHERICAL ASTRONOMY THE MOON. 428. The moon revolves in an elliptical orbit, of which one of the f oci is at the earth's centre. Its motion is from west to east, and its an- gular velocity about the earth is much greater than that of the earth around the sun. The moon appears, therefore, to move among the fixed stars in the same direction as the sun, but more rapidly ; and from the axial motion of the earth she has, like other heavenly bodies, an apparent diurnal motion, by which she rises in the. east, passes the meridian, and sets in the west. 429. The oblateness of the earth would be quite appreciable to an ob server at the distance of the moon. Her equatorial horizontal parallax is therefore found from Eq. (24) ; her distance from Eq. (28) ; her true diam- eter from Eq. (29); arid her mass from her effects in producing precession and nutation. Lunar Orbit. 430. The elements of the moon's orbit may be found from four ob- served right ascensions and declinations, corrected for refraction, parallax, and semi-diameter. Let D C be an arc in which F1 ? 82 - the plane of the orbit cuts the celestial sphere ; V B an arc of the ecliptic, and V A of the equinoctial ; V the vernal equi- nox, N the ascending node, P the perigee, and J/,, M. 2 , M*. MI the geocentric places of the moon. First convert the geocentric right ascensions and declina- tions into geocentric longitudes -md latitudes, and make v = V N :== longitude of node ; i =. C 'N B = inclination of orbit; li = V 0, = longitude o X, = J/i 0, a latitude of Jf, ; THE MOON. in then in the right-angled triangles Jt/j JV 0, and M 2 N 8 , we have sin (I, - v) = cot f . tan X, i sin (4 v) = cot * . tan X 2 ) And by division sin (1 L v) tan X, sin (1 2 v) tan X 2 * Adding unity to both members, reducing to common denominator, then subtracting each member from unity, reducing as before, and finally divi- ding one result by the other, we find sin (/ 2 v) -f- sin (/, v) tan X 2 -f- tan X, sin (7 2 v) sin (/, v) ~~ tan X 2 tan X, ' replacing the members by their equals, we have Also, from first of Eqs. (135), we have cot = sin( V-^ . (137) tan X, whence v and i are known. The longitude of the ascending node, increased by the angular dis- tance of a body from the same node, is called the Orbit Longitude, Make v, = YEN + NEM, = orbit longitude of M l ; p=VEN+NEP " " perigee;

I* ^> m Eq. (139), give four values for v, viz., v,, v,, i>,, and v 4 ; and these, and the times SPHERICAL ASTRONOMY. of observation h, t^ t$, and 4 , in Eq. (138), give four equations involving the four unknown quantities Z, m, e, and p ; whence these become known precisely as in 197, employing for the purpose Eqs. (50), (51), (53), and (54). 431. Denoting the ecliptic longitude VO of the perigee by p { , we have, in the triangle NP 0, right-angled at 0, tan N = tan (p v) . cos i, and Pi = v + tan" 1 [tan (p v) . cos i] . . . (140) 432. In the same way, denoting the mean ecliptic longitude of the moon at the epoch by L\, LI = v + tan' 1 [tan (L v) . cos i] . . . . (141) 433. The passage of the moon through one entire circuit of 360 around the earth, is called a sidereal revolution. The interval of time re- quired to perform a sidereal revolution is called a sidereal period. Denote the sidereal period by s, then will 360 * = - ......... (142) m The equation of the orbit, the centre of the earth being the pole, is I + e cos (v p)' and the value of r being found by means of Eq. (28), that of the mean distance a will result, and every thing in regard to the moon's path be- comes known. 434. At the epoch January 1st, 1801, the elements of the lunar orbit were Mean a = 59.96435000 of the earth's equatorial radius; " s = 27.321661418 mean solar days; " e= 0.054844200 ; _ " v= 13 53' 17".7; " ^, = 266 10' 07".5; " . = 508'47".9; " L l = 118 17' 08".3. g 435. The moon's true diameter, Eq. (29), is 0.27280, or about 2153 miles; volume, 0.0204; mass, 0.011399; density, 0.5657; and surface gravitation, 0.1666. 436. Comparing the lunar elements which depend upon the orbit aa THE MOON 113 determined at different times, they are all found to va,y. The nodes have a retrograde and the perigee a cfTrect motion, the former performing a com- plete revolution in 18.6, and the latter in 8.854 years. The inclination fluctuates between 4 57' 22" and 5 20' 06" ; the mean distance has a secular variation, and it is at the present time diminishing ; the same is true of the sidereal revolution, and the mean motion of the moon is increas- ing. All these changes are due to the disturbing action of the other bod- ies of the system, but principally of the sun. The action of the protuberant ring of matter about the equator of the earth also has its effect. Disturbing Forces. Fi &- 437. To illustrate the way in which these changes of the lunar orbit are brought about, let E be the earth, S the sun, M the moon, moving in her orbit in the &\vwt\oi\MDN'N', N and N' be- ing the nodes, and EV the direction of the vernal equinox. Then, resuming Aquations (80) and (81), making p = EM, the radius vector of the moon, and employing in all other respects the nota- tion of 286, v becomes the change which the sun's attraction causes in the weight of a unit of the moon's mass due to the earth's attraction, and r the change which the sun's attraction causes in the force normal to the radius vector and in the plane passing through the sun, earth, and moon. This latter force being in general oblique to the plane of the lu- nar orbit, urges the moon out of that plane, and causes her to describe a curve of double curvature, while the former has no such action. Resolve T into two components, one perpendicular to the radins vector and in the plane of the orbit, the other normal to this latter plane. For this purpose conceive a sphere of which the centre is at that of the earth, and radius, the radius vector p = EM, of the moon. Its surface will be cut by the plane of the ecliptic in AN t BN ti , by that of th* lunar orlit 8 SPHERICAL ASTRONOMY in N t MN tl , and by that of the sun, earth, and moon in AM B. Make O = V E S = sun's longitude ; ft = YEN, long, of moon's node; i = MN 4 A = inclination of lunar or- bit; X = N t MA = inclination of orbit to plane of sun, earth, and moon ; * = conjunction again, or from opposition to opposition, is- called a synodic revolution. Her passage from one longitude to the same longitude again, a tropical revolution ; from perigee to perigee, or from apogee to apogee, an anomalistic revolution ; from one node to the same node again, a nodi- cal revolution. The intervals of time required to perform these revolutions are called periods. 449. To find the length of either of these periods, say the synodic, let S be the sun, E the earth, M the moon in con- junction, E\ the place of the earth at the next conjunction of the moon, then at M { . Draw JStMi parallel to E S. At the sec- ond conjunction the moon will have re volved through 360 about the earth, in creased by the angle M^E\M\ E S E = the earth's angular motion in the same time. Make m = moon's mean daily motion ; = earth's " t = synodic period. Then tm = 360-f- E S E, r and by subtraction, Fig. 84. tn = 360; THE MOON. 360 whence < = - . . . . . . (146) in. -la. \ t m n 450. Here n denotes the real angular motion of the earth, which is equal to the apparent angular motion of the sun. If it be replaced bv the apparent geocentric motion of the vernal equinox, that of the apogee, or that of the node, taking care to give to each its appropriate sign (plus when the motion is direct and negative when retrograde), the correspond ing period will result. The mean daily motion of the vernal equinox is equal to 50".2 divided by 365 d .242-f ; that of apogee to 360, divided by the number of mean solar days in 8.854 years; and that of the node by the number of days in 18.6 years. The synodic period of moon = 29.53 -f- mean solar days. The anomalistic " " = 27.55 -f u tt The tropical u u = 27.32 -f " The nodical " = 27.21 + u The synodic period of the moon is called a lunar month, or lunation. Lunar Phases, 451. The sun's distance from the earth being 23984, and that of the moon only 59.96 times the earth's radius, the angle at the sun subtended by the semi-transverse axis of the lunar orbit is 08' 30"; so that rays of light proceeding from the sun to the moon and earth may be regarded as sensibly parallel; and the exterior angle of elongation S P E', Fig. 74, and Eq. (131), may be assumed equal to the true elongation SEP. Also the variation in the moon's distance is too small to produce sensible change in her apparent diameter to the naked eye, and the change be- comes perceptible only when viewed through measuring instrument*. The apparent diameter varies from 29' 21 ".91 to 33' 31".07, that at the mean distance being 31' 07". 452. Resuming Eq. (131), making d constant, and & equal to the moon's elongation, and supposing the sun to the right of the figure in the direction of E S produced, the earth at JS, and the moon successively in the positions 1, 2, 3, 4, 5, 6, 7, 8, we shall find the phases represented in the figure on next page. When in conjunction at 1, d or the elongation is zero, the moon is in- risible, and this phase is called new moon. When at 2, the elongation being 45 east, the moon is said to be in first octant, and the phas^ is SPHERICAL ASTRONOMY. Fig. 86. 0) crescent. When at 3, the elongation being 90 east, the moon is said to be in first quarter, and the phase is dichotomous. When at 4, the elon- gation being 135 east, the moon is said to be in second octant, and the phase is gibbous. When in opposition at 5, the elongation is 180, the phase is full, and is called full moon. When at 6, the elongation being 1 35 west, the moon is said to be in third octant, and the phase is gibbous. When at 7, the elongation being 90 west, the moon is said to be in the third quarter, and the phase is again dichotomous. When at 8, the elou Cation being 45 west, the moon is said to be in fourth octant^ and the .jhase is crescent. The interval of time required for the moon to pa^s through all these phases and resume them anew, is one synodic period, or lunation. 453. The earth presents to the moon the same phases that the moon does to us ; the angle of elongation of the earth, as seen from the moon, being always the supplement of the elongation of the moon, as seen from the earth. 454. The pale light of the moon, by which its outline is defined in conjunction, is due to the light reflected from the earth, then full, falling upon the dark side of the moon ECLIPSES OF THE SUN AND MOON. 455. The planets and sate. lites, being opaque, non-luminous bodies, and receiving their light from the sun, which is of vastly greater size, cast conical shadows, of which the surfaces produced must always be tangent to the sun's surface. The axes of the shadows cast by the planets, lie in the planes of their respective orbits. That of the earth is in the plane of the ecliptic ; and if at the time of syzygy the moon be near one of her nodes, she will either pass within the luminous portion of the conical ECLIPSES 119 space between- the earth and sun, or enter the earth's shadow, according as her phase is new or full. In the first case, she will mask the whole or part of the sun from some portions of the earth's surface ; and in the latter, will suffer a loss of the light she herself receives from that body. 456. The obscuration of the sun, by the interposition of the moon oetween the sun and earth, is called a solar eclipse. The obscuration of the moon, by a loss of solar illumination while within the earth's shadow, is called a lunar eclipse. Fig. 86. 457. Let S be the sun, E the earth, D Fj and C V^ tangents to the sun and earth ; A F t B will be the earth's shadow. Let M be the moon just entering the shadow, and H H 1 a right section of the latter at the distance of the moon. Make if = E C A = sun's horizontal parallax ; tf = C E S = sun's apparent semi-diameter ; P = E H A = moon's equatorial horizontal parallax ; s = HE M = moon's apparent semi-diameter ; R = E A = earth's equatorial radius ; then in the triangle E Fi (7, angle F, = tf if ; in the triangle E H F,, angle H= 180 P; and same triangle, E F, : EH : : sin (180 P) : sin (tf ir); whence sin (tf TT) (f if The least value for P is 52' 50" ; the greatest value for I ? Irt *"" J ^ to ^^ ot to 05 .s- d 05 S 2. i i 1 CO co 1 S ** to <* 10 05 V ^"a OS CO ^ "^ to t- .a OD C o d o 'R ei , o> to 4 - to s? CO 1 i Q a 3 ^ ^r to co CO 1 5 i J* i i O CO i i O5 C 1 'i O "S a> . ^2 O5 s II 01 GN 5 1 f 1 1 f 1 c- c 0> o i 1 CO 1 1 to 1

, would form an image of Jupiter's disk at F. An eye placed at E, within the field of the diverging waves, would re- ceive the light from this image and that from a star proceeding along the line BE. The ap- parent brightness of Jupiter's image would vary inversely as the square of FE, because this planet has no sensible phases, and under the same atmospheric circum- stances is of a constant brightness, while that of the star would be constant for all positions of the eye, and by altering the place of the latter the star and the image may be made to appear equally bright. The value of EF being ascertained for different stars, their relative brightness becomes known. 10 SPHERICAL ASTRONOMY. 546. Astronomers have generally agreed to restrict the first magni- tude to about 23 or 24 stars, the second to 50 or 60, the third to about 200, and so on, their numbers increasing rapidly as we proceed in the order of decreasing brightness, the number of stars registered to include the sev enth magnitude being from 12 to 15 thousand. 547. Stars of the first three or four magnitudes are distributed pretty uniformly over the celestial sphere, the number being somewhat greater, however, especially in the southern hemisphere, along a zone following the course of a great circle through the stars called s Orionis and a Ousis. But when the whole number visible to the naked eye are considered, they increase greatly towards the borders of the milky way. And if the tele- scopic stars be included, they will be foiind crowded beyond imagination along the entire extent of tliat remarkable belt and its branches. Indeed, its whole light is composed of stars of every magnitude from such as are visible to the naked eye to the smallest point perceptible through the bet 548. The general course of the milky way, neglecting occasional de- viations and following the greatest brightness, is that of a great circle in- clined to the equinoctial under an angle of 63, and cutting that circle in right ascension O h 47 m and 12 h 47 m , so that its northern and southern poles are respectively in right ascension 18 h 47 m and 6 h 47 m . 549. This great circle of the celestial sphere with which the general course of the milky way most nearly coincides, is called the gallactic circle. To count the number of stars of all magnitudes visible in a single field of a telescope, and to alter the field so as to take in successively the entire celestial sphere, is to gauge the heavens. 550. A comparison of many different gauges has given the average number of stars in a single field of 15' diameter, within zones encircling the poles of the gallactic circle, found in the following Table. Zones of North Gallactic Average Number ol Stars Polar distance. in field of 15'. to 15 . . . 4.32 15 to 30 . . . . 5.42 30 to 45 . . . . 8.21 45 to 60 . . . . 13.61 60 to 75 . . . 24.09 75 to 90 53.43 STARS. 14.7 Zones of South Gallactic Average Number of St*r Polar distance. it field of 15'. to 15 '. . . . 6.05 15 to 30 . . . . 6.62 30 to 45 . . . . 9.08 45 to 60 . . . . 13.49 60 to 75 . . . . 26.29 75 to 90 . . . . 59.06 551. This shows that the stars of our firmament, instead of being scattered in all directions indifferently through space, form a stratum of which the thickness is small in comparison with its length and breadth, and that our sun occupies a place somewhere about the middle of the thickness, and near the point where it subdivides into two prin- cipal laminae, inclined under a small angle to one another. For to an eye so situated, the apparent density of stars, supposing them pretty equally scattered through the space they occupy, would be least in the direction A S, perpendicular ;o the laminae, and greatest in that of its breadth S B, S C, or SD ; in- creasing rapidly in passing from one direction to the other. 552. For convenience of reference and of mapping, the stai's are sep- arated into groups by conceiving inclosing lines drawn upon the celestial sphere after the manner of geographical boundaries on the earth. The groups of stars within such boundaries are called constellations The brightest star in each constellation is designated by the first letter of the Greek alphabet, the next brightest by the second, and so on till this alpha- bet is exhausted, when recourse is had to the Roman alphabet, and then to numerals. A star will be known from the name of the constellation and the letter or numeral : thus, a Centauri, 61 Cygni. Many of the bright- est stars have also proper names, as Sirius, Arcturus, Polaris, ] Argus are very remarkable. In 1677 it appeared as a star of the fourth, in 1751 of the second, in 1811 and 1815 of the fourth, in 1822 and 1826 of the second, in 1827 of the first, and in 1837 of the second magnitude. All at once, in 1838, it sud* denly increased in lustre so as to surpass all the stars of the first magnitude except Sinus, Canopus, and a Centauri. Then it again diminished, but not STARS. 153 below the first magnitude, till April, 1843, when it had increased so as to surpass Canopus, and nearly equal Sirius. 573. On careful re-examination of the heavens, and comparison of catalogues, many stars are missing. 574. Double Stars. Many of the stars when examined through the telescope appear double, that is, to consist of two individuals close to- gether. They are divided into classes according to the proximity of their component individuals. The first class comprises those only of which the distance does not exceed 1" ; the second those in which it exceeds 1", but falls short of 2" ; the third those in which it ranges from 2" to 4" ; the fourth from 4" to 8" ; the fifth from 8" to 12" ; the sixth from 12" to 16" ; the seventh from 16" to 24" ; and the eighth from 24" to 32". Each of these classes is subdivided into two others, called respectively conspicuous and residuary double stars. The first comprehends those in which both individuals exceed the 8.25 magnitude, and are therefore sep- arately bright enough to be seen with telescopes of very moderate capa- city ; the second embraces those which are below this limit of visibility. Specimens of each class will be found in the following y Coronae Bor. y Centauri. y Lupi. c Arietis. t Herculis. Table. CLASS L 0" TO 1". n Coronae. 17 Herculis. A Cassiopeia?. A Ophiuchi. v Lupi. 7 Ophiuchi. p Draconis.

i CM CO uo uo 00 rHeoocoTfcocooo^uoococooocoojuocor^ uo t** CO uo uo UO ^ "Tf "^ CO CO 00 "^ CM t^* t"^ ^ CM CM "^ "^ w r-3 2 a ^ CM COCM'^CMCM^uoCMuOCMTPCMCOCMCOCM'HrM gg *5 H POSITION OF NODE. CO GOOOCMl > -CM' iCMCMUOCOO5T}- CM rH CM CM "^ uo ^ rH uo uo uo CO CM CM O3-*rHUO^OOUOt-tCOUOO5Tf-couoCMCMuOr-COrHO tt^-^COl^-COCOOCOCOO5COCOO50Ot^-OO5UOOO 1 000000000000000000000 1 " O500CM' v -OOl-t > -OOCMCMuOCOrHOCOOOOOO ^OC^O '^OCMODCM^OOCOfOiiCMUOOOUOOOCOOSrHCMic y ?rHrHrHP5COCMO'3 - circumstance connected with it indicates it to be F '^- ** a lenticularly-formed envelope surrounding the sun, and extending beyond the orbits of Mercury Ffrrizcii ^ and Venus and even to the Earth, its vertex having been seen 90 from the sun in a great circle. Different parts of the heavens furnish examples of similar forms. Figs. 25, 26, 27, Plate XIT. 604. Aerolites. Nothing prevents that the particles of this vast ma- terial envelope may have tangible size and be at great distances apart, and yet compared with the planets, so called, be but as dust floating in the sunbeam. It is an established fact that masses of stone and lumps ol iron, called Aerolites, do occasionally fall upon the earth fiom the upper regions of the atmosphere, and that they have done so since the earliest records. On the 26th April, 1803, one of these bodies fell in the imme- diate vicinity of the town of L'Aigle, in Normandy, and by its explosion into fragments, scattered thousands of stones over an area of thirty square miles. Four instances are recorded of persons having been killed by the descent of such bodies, and after every vain attempt to account for them as coming originally from the earth, and even from the moon, by volcanio projections, their planetary nature is now generally admitted. Their heat when fallen, the igneous phenomena which accompany them, their explo- sion on reaching the denber regions of our atmosphere, are accounted for by the condensation in front of them created by their enormous velocity, and bty- the relations of air, in a highly attenuated state, to heat. 605. Meteors. Besides these more solid bodies, others of much less density appear also to be circulating around the sun at the distance of the earth from that luminary. These on corning within the atmosphere ap- pear as shooting stars, followed by trains of light, '\nd are called Meteors. They appear now and then as great fiery balls, traversing the upper re- gions of the atmosphere, sometimes leaving long luminous trains behind them, sometimes bursting with a loud explosion, and sometimes becoming quietly extinct. Among these latter may be mentioned the remarkable meteor of August 18th, 1783, which traversed the whole of Europe, from Shetland to Rome, with a velocity of 30 miles a second, at a height of 50 miles above the earth, with a light greatly surpassing that of a full moon, and diameter quite half a mile. It changed its form visibly and quietly, separated into several distinct pprts, which proceeded in parallel direc- tions, each followed by a train. 606, On several occasions meteors have appeared 'n *tonishin 16J. SPHERICAL ASTRONOMY. cumbers, falling like a shower of rockets or flakes of SIKW, illuminating at once whole continents and oceans, even in both hemispheres. And it is significant that these displays have occurred between the 12th and 14th November and 9th and llth August. In November they are much more brilliant, but their returns less certain than in August, when numerous large and brilliant shooting-stars with trains are almost sure to be seen. 607. Annual periodicity, irrespective of geographical location, points at once to the place of the earth in its orbit as a necessary concomitant, and leads to the conclusion that at that place the earth enters a stratum, or annular stream of meteoric planets, in their progress of circulation around the sun. The earth plunging in its annual course into a ring of these bodies, and of such thickness as to be traversed in a day or two, their motions, referred to the earth as at rest, would be sensibly uniform, recti- linear, and parallel. Viewed from the centre of the earth, or from any point on its surface, neglecting the diurnal as being insignificant in com- parison with the annual motion, their paths wojild appear to diverge from a common point on the celestial sphere. Now this is precisely what haj>- pens. The vast majority of the November meteors appear to describe arcs of great circles passing through y Leonis, and those of August appear to move along paths having a common point in /3 Camelopardi. 608 As the ring may have any position and be of an elliptical fig- ure having any reasonable eccentricity, both the velocity and direction of denotes the mean longitude of the moon's node, D the true longitude of the moon, and the longitude of the sun. 639. And assuming the mean obliquity of the ecliptic for 1850 equal to w = 23 27' 31 ", we have then for the nutation in longitude, denoted by J //, A L - 17".3017 sin & + 0".20S1 sin 2 & - 0".2074 sin 2 D 1".2552 sin 2 CATALOGUE OF STARS J^J 640. Denoting the equation of the equinoxes in right ascension by J A, we have A A = 15". 872 sin ft -f 0".192 S.TI 2 ft 0".190 sin 2 }) 1".500 sin 2 0. 641. Denoting the right ascension and declination of any body by a and 5 respectively, and by p and p', its change in the same due to an- nual precession, then will p = 46".05910-f 20".05472 sin a .tan d . . . (168) p' = 20".05472 cos a (169) 642. The change in right ascension and declination for any fractional portion of the year will be found by multiplying the above by ' = 365^25 = - 002 ' 73785 X d ' ' ' ' ( 170 > In which d denotes the number of days from the beginning of the year to the end of the fraction. 643. Denoting by da., and d8 t the change in right ascension and dec- lination arising from nutation, then, omitting terms involving sin 2 D , will L-. a, = (15".872 + 6".888 sin a . tan i) . sin ft '9".250 cos a . tan (5 . cos ft ) f (o".191-f 0".083 sin a . tan I) . sin 2ft+.0".090 cos a . tan 6. cos 2ft V (171) (1".151+0".SOO . sin a . tan t) . sin 2 0".545 cos a . tan S . cos 2 \ A d, = 9". 250 . sin a . cos Q 6 ".888 cos a . sin ft J - 0".090 sin a cos 2 ft + 0".083 cos a . sin 2 ft [ . (1Y2) + 0".545 sin a . cos 2 0".500 cos a . sin 2 O 5 g 644. Aberration. Denoting by ^ 8 and 4 6 a the change in right ascension and declination arising from aberration, disregarding the eccen- tricity of the earth's orbit, A a 2 s-s - (20".4200 sin . sin a -f 18".7322 cos Q cos a) . sec I . . (178) A<5 2 = (20' .4200 sin Q . cos a - 18".7322 cos O sin a) sin i \ 8".1289 cos cos i ) 645. Multiplying Eq. (168) by Eq. (170), adding together the prod- uct and equations (171) and (173), and denoting the apparent right as- cension by a and the mean by a', there will result, after suitable reduction, a' a = Aa = ( 0.848 rfn fa + 0.004 sin 2 ^ 0.026 sia 2 ) X (46".069 -f 20".OK> sin a tan to - (9".260 cos & - 0''.090 cos 2 ft + 0".545 cos 2 ) . cos a . tan i 20".420 sin . sin a . sec i 18".732 cos Q . cos a . sec 6 0".0530 sin ft -f 0".000 sin 2 ft 0".0039 sin 2 Q. SPHERICAL ASTRONOMY. Mu/tiplying Eq. (169) by Eq. (170), adding together the product and equations (172) and (174), and denoting the apparent declination by d and the mean by a = cos a . sec , b = sin a . sec , c 46".059 + 20".055 sin a . tan 6, d = cos a tan , a' = tan w . cos sin a . sin 5, &' = cos a . tan , c' = 20".055 cos a, d' = sin a ; the above become d. .... (173) d'.D .... (176) 646. Proper Motion. To the foregoing must be added the proper motion of the star when it is known with sufficient accuracy, and is of sufficient magnitude to be taken into the account. Equations (173) and (174) enable us to pass from the apparent to the true, or from the true to the apparent right ascension and declination of a star. 647. Since the motion of the equinoxes is very slow, the values of the functions a, 6, c, d, a', b', c', and d' will be sensibly constant for a number of years, particularly when the stars are not very near the poles, while those of the functions J, B, C, and D vary sensibly from day to day These latter are, therefore, computed for every day in the year, and their logarithms recorded in the astronomical ephemeris ; the others are com- puted for the epoch of the catalogue, and their logarithms recorded oppo site each star in tbe catalogue. CATALOGUE Of STARS. 173 g 64 g. Construction of the Catalogue. The elements relating to each star occupy a portion of the two pages exposed to view on opening the catalogue. On the left-hand page will be found every thing relating to right ascension, and on the right, to declination. The left-hand page con- sists of eleven vertical columns : in the first is placed the number of the star, in the order of its right ascension ; in the second, the name of the con- stellation in which it is situated, with its letter or number; in the third, its magnitude; in the fourth, its mean right ascension, January 1st, 1850, in time ; in the fifth, its mean annual precession in right ascension, Eq. (168), reduced to time ; in the sixth, its secular variation, reduced to time ; in the seventh, its proper motion in right ascension, reduced to time; and in the eighth, ninth, tenth, and eleventh, the logarithms of the functions a, 6, c, and d, reduced to time, respectively, each preceded by the sign of the function to which it belongs. The right-hand page consists of fifteen vertical columns, in the first of which the number of the star is repeated ; the second contains the mean north polar distance, January 1st, 1850; the third, fourth, and fifth, the annual precession, secular variation, and proper motion in north polar distance, respectively ; the sixth, seventh, eighth, and ninth, the logarithms of the functions a', &', c\ and d' re- spectively, each preceded by the sign of the function to which it be- longs ; the remaining columns contain the numbers by which the star is recognized in the catalogues of the several authors, whose names are at the top. Example. Required the apparent right ascension and declination oi y Orionis, February 5th, 1854. Mean a January 1st, 1850 . 4 years' prec. and pr. motion Mean a January 1st, 1854 . b. m. . o " 5 17 05.33 Mean N. P. D. . . . 83 47 25.7 -f- 1 2.88 4 y'ra' prec. and pr. motion 14.9 5 17 18.21 Mean N. P. D. . . . . 83 47 10.8 a A a A b B bB Logs. -f 8.0963 1.1363 - 9.2326 -f 8.8188 -f 1.1443 9.9631 Nat No. -OM71 -f 0.919 A of A 9 B VB Nat. No* -f- 0.6483 . -f- 4' .449 8.3039 + 1.1443 - 9.4482 - 0.281 174 SPHERICAL ASTRONOMY. Logs. Nat. Nos. Logs. Nat c . -f 0.5070 c' . 0.5721 C 9.2812 C 9.2812 e C . - 9.7882 - 0.614 c' C . + 9.8533 . -f 0.713 d . + 7.1304 d' . + 9.9923 D . 0.5713 I> . 0.5713 7.7017 0.005 d'D . 0.5636 . 3.661 A a =+0.129 AN. P. D. = + 1.220 Hence app't, right ascensk n, Feb. 5, 1854, 5 h I7 m 18-.21 -f 8 .13 = 5 h 17 ra 18 8 .34 app't N. P. D 83 47' 10".80-f 1".22 = 83 47' 12".02 APPLICATIONS. TIME OF CONJUNCTION AND OF OPPOSITION. 649. To find from the ephemeris the time at which two bodies are in conjunction or opposition, find by inspection two simultaneous longitudes, one for each body, that differ by or 180. The corresponding time of the first will be that of conjunction, and of the second of opposition. 650. But if these longitudes are not to be found in the tables, take therefrom two consecutive longitudes for each body, such, that those of the first shall differ from those of the second, in order, the leavSt possible. Then, denoting the lesser and greater longitudes of the body having the greater velocity by V and /'', those of the other by l t and l n respectively, and the corresponding times by t' and rf", we have, because the longitudes of each are given for the same epochs, (I" _ /') - (/ - i t ) : (t " - t ') : : I, - V : *, whence m which x denotes the interval of time from t f to conjunction. And de- noting the ephemeris time of conjunction by T<,, we have 651. Increasing thr longitudes of one of the bodies by 180, and se- PROJECTION OF A SOLAR ECLIPSE. 175 lecting those of the other to differ the least possible from these increased longitudes, then will T e become the time of opposition. 652. T c is the local time on the meridian for which the ephemeris is computed. Denoting the longitude, in time, of any other meridian west of this one by Z-, and the local time of conjunction or opposition by T. then will T=T C -L (178) ANGLE OF POSITION. 653. The angle made by a circle of latitude with a circle of declination through the centre of a body, is called the angle of the body's position. To find this angle, let P be the pole of the ecliptic, P / that of the equinoctial, and S the cen- tre of the body, and make X = 90 P S = latitude of the body ; S = 90- P / S = declination of the body ; a = P P' obliquity of the ecliptic ; S = P S P' = angle of position : then will and Fig. 101. cos rt = sin X . sin 8 + cos X . cos 8 . cos S, cos TX sin X . sin 8 cos S = cos X . cos 8 654. If the body be the sun, then will X = 0, and cos* cos S cos 8 (179) (180) PROJECTION OF A SOLAR ECLIPSE. 655. A solar eclipse can take place only at new moon. Find the aphemeris time of the moon's conjunction with the sun. Then, by the method of interpolation, determine the sun's true longitude and hourly motion in longitude ; the moon's true longitude and latitude, and hourly motion in longitude and latitude ; the sun's and moon's horizontal paral- laxes, and apparent semi>diameters, and the sun's angle of position. 656. Conceive a cone tangent to the earth, and of which the vertex is at the sun. A section of this cone, by a plane between the earth and sun, will give an area upon which the sun's centre will appear tc be pro- 176 SPHERICAL ASTRONOMY. jected when viewed from different parts of the earth. A section at the distance of the moon from the earth, and perpendicular to the axis, is called the circle of projection. 657. The diurnal rotation of the earth carries an observer once around his parallel of latitude in 24 hours ; a line connecting him with the centre of the sun, describes an entire conical surface in the sam^ time, and a sec- tion of this cone by the circle of projection will be the paral lactic path of the sun as determined by the axial motion of the earth. This ellipse and the relative orbit of the moon, with a scale of time on each, indicating the simultaneous positions of the sun and moon, being constructed upon the plane of projection, all the circumstances of a local solar eclipse may easily be predicted. Fig. 102. 658. Sun's Parallactic Path. -Let P G P'H be a meridian section of the earth by a declination circle through the sun's centre at S ; E the earth's centre ; P the elevated pole ; Gr H the projection of the equator ; B A that of the observer's parallel, and N N' that of the circle of projec- tion on the plane of the section. The projection A'B', of A B on the circle of projection by the lines A S and B S, will be the conjugate, and that of the diameter of the parallel, which is perpendicular to A B, the transverse axis of the ellipse ; the first being in and the second perpendic- ular to the declination circle through the sun. 659. Make P = E N'U'= moon's horizontal parallax ; = E S U'= sun's / = A E H = reduced latitude of place ; d ffJS = sun's declination ; p = E A = earth's radius ; u = number of seconds in radius. PROJECTION OF A SOLAR ECLIPSE. Draw AC, ED, and ^^perpendicular to E S, and we A C= p .sin (I - d) ; B D = p . sin (I + d) ; ^ (7 = p . cos (I - d) ; E D = p . cos (I + d). Also, Eq. (28), 177 ' whence From the figure, P * .__. S C= ES EC=? . p . cos (/ o?). Then in the triangles S C A and S MA', SC : SM :: AC : A'M, and by substitution rin(/-d) ?.-* also, S D = 1 _ I . cos (/ - d) = p . - + p cos (/ + and in the same way as above, from the triangles S D B and S M B', sin (l + d) P- But ie can never exceed 9", and u is equal to 206264".8, so that the terms into which *r -7- w enters as a factor may be neglected, and we have n (l- d ).~- (181) JT Q n + d).^- ....'. (182) MA f = From which we see that the length of the projection of any dimension at the earth, and parallel to the circle of projection, is found by multiply- ing this dimension by (P t) -r- P. 660. Denoting the conjugate axis A'B' by 2 6, we have = M&- MA', 12 SPHERICAL ASTRONOMY. and by substitution, Also, b = p . cos / . sin d . P - (183) FA = p . cos I ; and because that diameter of the parallel of latitude, which is perpendic- ular to A B, is parallel to the plane of projection, we have, denoting the semi-transverse axis of the ellipse by a, P a = p . cos I . (184) And denoting the distance M F r from the centre of the circle of projec- tion to that of the ellipse by JT, we have, taking half sum of equations (181) and (182) Y = p > . sin I . cos d . (185) 661. Revolve the parallel of latitude about AE till it coincides with the meridian section. When the observer is at A, it is to him apparent noon ; when at B, apparent midnight; when at 0, the angle OF A is the apparent hour angle of the sun, and therefore local apparent time. Fig. 102 bis. Draw T perpendicular to A B, and S L through the point T. The projection of F L will give the distance of the sun from the transverse, and that of This distance from the conjugate axis of his elliptical path. Denote the first by y, the second by ar, and the hour angle F A by &. -jThen FO = FA = ? .coal-, T = p . cos I . sin h ; FT = p. cos /.cos h\ PROJECTION OF A SOLAR ECLIPSE. 179 and since F L T is sensibly a right angle, the value of E S U, which is much greater than E S T, never exceeding 9" ; and because FTL = UEP = d, we have F L = F T. sin d p . cos I . cos h . sin d ; and projecting F L and T on the circl 3 of projection, there will result . . , , p-r y = p . cos / . sin d . cos h . - x = p . cos I . sin h . P- (186) (187) But p -r- P is the linear subtense of the unit of arc in which P is ex- pressed say one minute. Calling this distance unity, equations (183), (184), (185), (186), and (187) may be written b = cos I. sin d (P' ') (188) a= P' + cos \P> sin \r P' ; making cos -J r' P = 1, squaring and rejecting the term containing the second power of sin J r' P', we find TERRESTRIAL LATITUDE. 195 sin a JP = sin 2 P' + 2 sin \ P' cos i P' . sin \ r' P' ; but 2 sin -JP'. cos JP' = sin P', and since P' and r' are both small, sin P' = 2 sin P', sin i r' P' = r f sin 1 P 7 ; which substituted above give sin 2 P = sin 2 A P' + 2 r' sin 2 J- P' = (1 + 2 r') sin 2 P' ; and finally making i = 1 + 2 r' = 1 + 0.000022 r . . . . (222) and substituting in Eq. (220) we have , cos I . cos d /cos / . cos dV x = i.Jc. =r 2 .w.cot (ld) .(-777 7-1 (223) sin (ld) \siu(l d)J m which it will be recollected that r, in the value of ', is the rate of the time-keeper, minus when the latter gains and plus when it loses on 'side- real time. 700. The first term in the second member of Eq. (223) will always ">e sufficient when the observations are made within five or ten minutes of .he meridian. And it is important to remark, in view of the use presently t,> be made of the value of X 9 that the latter will not be sensibly affected by a small error in the value of /, and that an approximate latitude may therefore be substituted therefor. The values of k and m are computed for all values of P'from to 35 m , and inserted in Tables V. and VI. TERRESTRIAL LATITUDE AND LONGITUDE. 701. The determinations of terrestrial latitude and longitude by , means of astronomical observations and ephemerides, are among the most important of the objects of practical astronomy. All appreciate the value of these determinations in navigation and geography, and we now proceed to consider them in the order named. Terrestrial Latitude. 702. The zenith distance of the pole is always the complement of the latitude of the place, and when known the latitude is known from the relation X ^ 90 L 106 SPHERICAL ASTRONOMY. in which X denotes the zenith distance " g '_ of the pole, and I the latitude of the place. 703. The zenith distance of the pole forms one side Z P of a spherical triangle, of which the two other sides, Z S and P S, form, respectively, the zenith and polar distances of some heavenly body, of which the angle at the pole is the hour angle, or distance of the body from the meridian. And the determination of latitude consists in the solu- tion of this triangle, the data for this purpose being the true zenith distance Z S determined from observation, the polar distance P S found from the ephemeris, and the hour angle Z P S, which is always equal to the sidereal time of observation, diminished by the body's right ascension at the same instant. Having, then, found the true zenith distance by cor- recting the observed for refraction, parallax, and semi-diameter when ne- cessary, and the body's true hour angle and polar distance from the time of observation, the ordinary formulas for the solution of spherical triangles will do the rest. 704. Latitude by Meridian Zenith Distance of a Body. But it is desirable, in practice, to select those moments for observations which will give most accurate results, and these are when the hour angle is or 180 ; in other words, when the body is on or near the meridian, for then it has the least change in zenith distance for a given interval of time. Make z = Z S = true zenith distance of body ; d = 90 P S = the body's declination ; P = Z P S hour angle of the body ; A = P Z S 180 the body's azimuthal angle. Then in the triangle Z P S, cos 2 = sin / . sin d -f- cos / . cos d . cos P . . . (224) sin d = sin I . cos -f- cos / . sin z . cos A . . . (225) 705. Making P = 0, the body will be on the meridian some- where between the poles on the side of the zenith, and A will be or 180. In the first case, the body will be between the zenith and elevated role cos A I, and Eq. (225) will become wuence and TERRESTRIAL LATITUDE. 197 sin d = sin / . cos z + cos I . sin z = sin (/ -f z) , ** + , J = d z (226) Flf. 118. Fig. 114. Iii the second case, the body will be on the opposite side of the zenith from the elevated pole, cos A = 1; and if the latitude and declination be of the same name, sin d and sin I will have the same sign, and Eq. (225) gives gin d = sin / . cos z cos / sin z = sin (/ z) ; whence d = / - z, and 2 = d + z (227) Fig. 115. If, in the second case, the declination and latitude be not of same name, the body will be below the equinoctial ; sin d and sin / will have contrary signs, and Eq. (225) gives whence and sin ( d) = sin I . cos z cos I . sin z sin (/ z); l-t-d (228) 198 SPHERICAL ASTRONOMY. If P = 180, the body will be on the meridian below the elevated pole, and A = ; cos P = - 1, and, Eq. (224), cos = sin /. . sin d cos I . cos e?= cos (/-f-d); whence Fig.im and 1= 180 -z + d (229) 706. Latitude by Circum-meridian Altitudes. Thus it is easy to find the latitude when the meridian zenith distance and declination of a heav enly body are known. The declination is found from the ephemeris, if the body belong to the solar system, or from the catalogue, if it be a star. The meridian zenith distance is best determined by the method of circum-meridian altitudes, which consists in measuring with an instrument a number of altitudes of the body just before and after its meridian pas- sage, noting the corresponding times ; reducing to the meridian, taking an average value of the results, and subtracting this from 90. 707. Denote by A,, A 2 , A 3 , &c., the measured altitudes ; r } , r s , r 3 , &c., the corresponding refractions; p h p. 2 , p 3 , &c., the parallaxes; A the ap- parent semi-diameter ; x lt x a , x 3 , &c., the reductions to the meridian ; n the number of observations ; and IT the average meridian altitude; then will r, -f ff t -f &c. (230) the upper sign corresponding to the lower limb, and vice versa. Denote by P,, P 2 , P 3 , &c., the watch hour angler of the body ; that is, the differ- ence between the watch time of meridian passage and those of observa- tions. These, with tables, give & & 2 , & 3 , &c., m,, ra 2 , ra 3 , ]>. sin h = cos A . sin (h 40 + s ^ n A cos (^ ~~ 40 cos P dividing by sin h and factoring, 1 =r cos ^ . (cos A -J- sin A cot h . cos P) sin L = l -i^A 2 -^^ 3 , sin 4, = A A -f B A 2 + (C- l^ 3 ) A 8 . Substituting the values of a, 6, cos 4^, and sin 4/, in Eq. (235), we have the identical equations, cot h . cos P A . cot h = 0, -%(l+A*) + A cos P ,8 cot h = 0, i A - % ( 1 + 3 A*) cos P - ( C - | ^4 3 ) = 0. Whence A = cos P ; ^ = % sin 2 P . tan h ; (7 = i cos P . sin 2 P ; which in Eq. (236) give 4, = A . cos P I sin 2 P . tan h . A 2 + cos P . sin IP . A 8 . To express -^ and A in seconds, write -^ s i Q 1" f r 4' an( ^ ^ sm 1" f r A, and make m = % sin 1", tt = sin 8 l", then will 4, = A cos P m (A . sin P) 2 . tan h + n . (A . cos P) . (A . sin P) 8 (237) This value applied with its proper sign to the observed altitude, cor- rected for refraction, will give the latitude. It is best to take some half dozen altitudes, and to note the corresponding times' in pretty rapid suc- cession ; a mean of the altitudes corrected for refraction will give A, and a mean of the sidereal times diminished by the right ascension of the star, and the remainder multiplied by 15, will give P. 711. This method is of such practical utility as to have caused the insertion into *he English Astronomical Ephemeris and Nautical Almanac of three tables, of which the first contains the value of A cos P for every 10 minutes, sidereal time, for a mean and constant value of A; the second contains the values of m . (A . sin P) 2 . tan h ; and the third contains 202 SPHERICAL ASTRONOMY. corrections to be applied to the values in the second tible. The secona and third tables are arranged in the form of double entry, the arguments for the former being the sidereal time and altitude, and in the latter side- real time and date. The third term of Eq. (237) is neglected as being insignificant. Longitude. 712. The longitude of a place is the angle made by its meridian with some assumed meridian taken as an origin of reference. The problem ot longitude is much more complex than that of latitude, and its solution consists, as we have seen, 94, in finding the difference of local times that exist simultaneously on the required and first meridian. 713. Longitude by Chronometers. Could the motion of a time-piece be made perfectly uniform, and the angular velocity of its hour-hand equal to that of the earth's axial rotation, without the risk of variation, the de- termination of longitude would be a simple matter. It would then only be necessary to put the time-keeper in motion ; on a given meridian ascer- tain, by the methods explained, its error on the local time of this meridian ; transport it to the unknown meridian, determine its error on local time there, and take the difference of these errors ; this difference would be the difference of longitude of the meridians in time. But such time-pieces cannot be made. The results to which they would lead may, however, be approached within limits all-sufficient for practical purposes. It is only necessary that the time-keeper shall run uniformly, a condition which chronometers have been made so nearly to attain as to vary their rate but half a second in 31536000 seconds. 714. By daily observations find the error of a chronometer ; from the variation of the error during the intervals between the observations, find that for 24 chronometer hours. This will be the rate. Make e= error on local time on gwen meridian, at some given epoch; plus when too slow, minus when too fast ; e = error on local time on required meridian, at some subsequent epoch ; e t =. error on local time on given meridian, at this hist epoch ; r rate; minus when gaining, plus when losing; i = interval of chronometer time between the .epochs at which e and e f are found always plus ; I = difference of longitude. Then / = -; . ./ TERRESTRIAL LONGITUDE. 203 Fig. 118. whence / = e, e + i . r (238) 715. Longitude by Lunar Distances. The moon has a rapid motion in longitude. Her geocentric angular distances from the sun, planets, and fixed stars that lie in and about her path through the heavens, are com- puted in advance and inserted into the Nautical Almanac. From these hours and distances is readily found, by interpolation, the Greenwich time corresponding to any given distance not in the Almanac, and the difference between this interpolated time and the local time on any other meridian at which the moon is found from observation to have this given distance, is the longitude of the meridian on which the observation is made. 716. Measure the altitude of the star, and that of the upper or lower bright limb of the moon ; also measure the angular distance from the star to the bright limb of the moon, and note the local time of this measure- ment ; correct the altitude of the limb and measured distance for semi- diameter ; then correct the altitude of the star for refraction, and that of the moon for refraction and parallax. Let Z be the zenith, Z S and Z M the arcs of vertical circles, the first passing through the star S and the second through the moon's cen- tre M. The effect of refraction being to ele- vate and that of parallax to depress, and the parallax of the moon being always greater than her refraction, the star will appear at S' above its true place, and the moon at M f below her true place. Make h 90 Z M' = observed altitude of moon's limb corrected for semi-diameter ; h' 90 Z S' = observed altitude of star ; d r = M' S r = observed distance corrected for semi-diameter of the moon ; H = 90 Z M = true altitude of moon's centre ; H' = 90 Z S = true altitude of star ; 4 = M S = true or geocentric distance between the moon's centre and the star ; z = MZS = angle at Z. Then in the triangle M'Z S', cos A' sin h . sin h' cos z = cos h , 2os h' 2Q4- SPHERICAL ASTRONOMY. and in triangle M Z S, cos A sin H . sin H' cos z = ~, ; cos H . cos If' equating these values of cos z, cos A' sin h . sin h 1 cos A sin H . sin IT cos h . cos h' cos H . cos H' adding unity to both members and reducing, cos A' + cos (h + h') _ cos A + cos (H + H') ^ cos h . cos h' cos If . cos If' Make A + A' + A' = 2 ra (239; whence cos (h -h h 1 ) = cos (2 m A') ; substituting this above and reducing, we find , H+H' 2 A , ,. cos j . sin 2 cos m . cos (m A ) 2 2 cos h . cos A' cos H . cos ^P whence sin 1 A = '/cos 2 J (J5T + T) - cos ^- cos f r . cos m . cos (m - A'), ' cos h . cos A' and making, to adapt the foregoing to logarithmic computation, . COS : rr cos m . cos (m A') cos A . cos A . . (240) then will result sin l A = cos i (H + If') . cos 9 . . . . (241) 717. The quantities A, A', H, H', and A', are obtained from observa- tions, and the corrections for semi-diameter, refraction, and parallax applied thereto ; the value of m is given by Eq. (239) ; the auxiliary arc

* 3 .... (251) which would be the increment of the function for an increment ot t equal to unity, were the function to increase uniformly and at the rate it had for any arbitrary value for t t . 728. 2. Observations.' Make . W. Limb 7 h 27" 47*, 66 f Cancri . . . . 8 o3 21 ,44 3)22 09 i3 , 21 7 23 04 , 4o a = A = fhen, Eq. (246), E I2&oooo, a = oo 10 34 , 53 Nautical Almanac c a = oo 25 41 , 18 t = 4 56 28, Log . . 4,6354837 " . . 2 , 8024620 " a. c. 6 , 8121918 . . 4, 2601275 Next, interpolate change of right ascension for I ; . 4*> 56 28' o i5 17', 8a o , o3 o 04 43 , 26 o' a io">34 s , 53 / = j , 12 oo oo, t . . . . Log . . 4, 2501275 " a.c. 5,3645i63 " . . 9,6146436 Nautical Almanac. Feb. 17, L. C. 7 h oi56', 27 " 18, U. C. 7 27 47,66 L. C. 7 53 28 , 84 19, U.C. 8 18 5 9 ,56 255i s , 39 25 41 , 18 25 30,72 (-10*, 21 .J / 10 , 46 A = 254i 8 , 18 + o5, 17 - o, 02 = 25" 46', 33 B- o5,T 7 -foo,o6 = o5,n C- ...... . - 00.04 Then, Eq. (249), A . . Log . . 3 , 1893022 t . . u . . 9 , 6i46438 2 , 8039400 Nos . . 63s, 72 B . . Log . . o , 7084209 F . . u . . 9 , 2292876 9, 9 377o85 Nos . . o , 87 . . Log . . 8 , 6190933 * . . . . 8,843 9 3i4 ^ , 4630247 Nos . . o , oo3 * .... 635,85 io 34 s , 53 = o .... 634,53 Again, Eq. (251), A Nos . . s546, 33 B . . Log . . "o , 7084209 t . . . . 9,6146438 8 . . . . o , 3oio3oo o , 6240947 Nos C . . Log . . 8,6190933 * . . " . . 9 , 2292876 3 . " o,477' 2 ' 3 8, 3255022 Nos . . - 0,02 = . . . . s5> 4a% 10 Then, last term of Eq. (247), H . . Log . . 4,6354837 a -a, . . " . . o", 1205739 v . . " a. c. 6,8118875 = - 36% 97 . . " . . T , 5679451 No* . . oo 36 , 97 Kq. (247), L = 4 h 56 m 28' - 36, 97 = 4 h 55" 5r, o3 212 SPHERICAL ASTRONOMY. 729. It frequently happens that the moon cannot be observed on the middle wire, in which case she is far enough from the meridian to have a sensible parallax in right ascension ; and as it may be very desirable not to lose the observation, this parallax must be computed and applied to the apparent hour angle from the middle wire, which is supposed to be nearly coincident with the meridian. Denoting the hour angle by A, the parallax in hour angle by A A, th,-. geocentric latitude by /, the moon's declination by Z>, and her horizontal parallax by P, then, Appendix XI, p. 379, A h p . cos / . sin P . sin h . sec D ; and^to make this applicable to the case before us, A will denote the equa- torial interval, in sidereal time, from the lateral to the central wire. This angle being small, its arc, expressed in seconds of time, may be taken for its sine, in which case, A A will be in time-seconds, and the true distance of the moon's limb from the central wire, denoted by A /? will be h, == A . (1 p . cos /.sin P . sec D) ; ind the reduction to the meridian, denoted by r, in time-seconds, A 1 p . cos I . sin P . sec D cos~7> ' ~ 1 0,04166 .m in which m is the moon's daily motion in right ascension in hours. The upper sign, when the observation is before the middle wire. The quantities p and I are found from tables on pp. 336, 337. 730. It also often happens that two observers do not use the same number of wires, or if they do, that the same stars are not observed at the same number. Such observations are not of equal weight. To rind the relative value with which such observations should enter into the tinal determination, Professor Gauss has given the following formula, deduced from the principle of least squares. Let the number of wires on which the moon is observed at one place be denoted by n t and at the other by n 1 '; and let the number of wires at which the stars are observed at the first place be a, b, c, &c., and at the other be a', &', c', & 3. Make r = X (253) n + _ a _^_ = a _A^_ = R -^L = r , fee. (254) a + a' b + \i c + c tf = a + + y + &r. (255) TERRESTRIAL LONGITUDE. 213 Then, if W denote the weight of each day's comparison, will in which z is the same as - in Eq. (247) ; and for the weight of the result of all the comparisons, we have ....... (257) in which 2 expresses the sum. Let e denote the probable error of observation, and E the probable error of the final result, then will ..... ( 258 > 731. Longitude by Telegraph. One of the simplest and most accurate methods for finding differences of longitude, is to telegraph to a western, the instant of a fixed star's culmination at an eastern station, and, conversely, to telegraph to the eastern the instant of culmination of the same star at the western station. The local times of both events being noted, the difference, as recorded at the same station, corrected for rate of time-keeper, gives the difference of longitude. The instant of culmination of the moon's bright limb being also sig- nalized in the same way, the difference of time, as recorded at the same station, corrected for rate, as before, gives the difference of longitude augmented by the limb's change in right ascension during the interval, and the excess of this interval over that for the fixed stars is the change itself. Thus the telegraph, where it connects stations remote froir one another, gives the means for finding differences of longitude and for correcting the lunar ephemeris, and, therefore, the elements employed in the method of lunar culminations, for use at stations having no telegraphic connections. 732. Longitude by Solar Eclipse, or by Occupation. The follow- ing elegant and accurate solution of this most important problem is, in substance, due to Mr. Woolhouse ; it first appeared in the Nautical Almanac for 1837. 214: SPHERICAL ASTRONOMY. Let M and S, be the moon and sun, in such geocentric positions as to appear in external tangential contact to an observer on the earth's surface ; the local time of this observer will be that of beginning or ending of the local eclipse^ Conceive a fictitious sun, s, at the distance of the moon, within and tangent to the visual cone that projects the true sun on the celestial sphere for this observer. This fictitious sun will be in con- tact with the moon ; and any parallactic effect on the one, due to a change in the observer's place, will be equal to that on the other. Transport the observer to the centre of the earth; the moon and fictitious sun will appear to shift their places with respect to the true sun; but, being in actual, will remain in apparent contact. The apparent disk of the fictitious sun and of the moon will diminish ; and the size and place of the latter will become those of the ephemeris at the instant of observation. The change of the fictitious sun's place, in reference to that of the trne sun, will be the effect of relative parallax. Apply this parallax to the place of the true sun, and diminish his disk by a quantity equal to the diminution of the fictitious sun ; the result will be the place and size of the latter body in apparent contact with the moon, to the observer at the central station. The ephemeris time of this contact, diminished by the local time of observation, will give the longitude of the observer. Thus, the determination of terrestrial longitude, by a solar eclipse, is reduced to finding the ephemeris time when the true disk of tlie moon comes in contact with a disk of a given size, placed at a given place. The principle is the same for an occultation of a star by the moon. In the case of a solar eclipse, the apparent time of observation, converted into arc, gives the hour angle of the sun's centre at that instant ; and, as the declination of the sun is never subject to a very rapid daily variation, this element may be taken from the ephemeris, with sufficient accuracy, for the approximate local time on the meridian for which the ephemeris is constructed, deduced from an estimated longitude, or rough longitude, by account. Take a = right ascension, h = hour angle, }. of true sun; 6 declination, a apparent semi-diameter, TERRESTRIAL LONGITUDE. 215 Take also a = right ascension, "| #o = declination, V of fictitious sun, tfo = apparent semi-diameter, J Act = Ah relative parallax of moon in right ascension, Ad = " " declination, Arf = diminution of fictitious sun's semi-diameter;. Then will a = a -f- &h, x x J A* FIg> m o = o -{- Ao, tf = , and semi-diameter tf ; the horizontal parallaxes P, if ; also take out the moon's declination J) roughly to the minute. Reduce the latitude by the table on p. 336, and with p from the table on p. 337, Ap. XI, find h = apparent time of observation reduced into arc. P = P' cos / sin A ; 4 h in min. = [7.92082] ^-yr; (h) = h J cosD v ' tan & = cos (h) cot I ; G = cos (h) cos I ; tan g tan (b + #) cos Jlf ; sn tan M - - -- - tan (A) ; cos (6 + 6) check 5 = cos Jf cos s ; sin ' cos(d + <5) A a in time = [8.82391] cos M to be in the same semicircle with k. 3. With s find the corresponding factor f in the annexed table ; then, using P and , hourly var. (D) hourly var. (<5) in arc ; K) = (a) + Aa; (S ) =($) + A 6. m = ( a ) ~ ( J ) ; t = (t) + m [3.55630] ; -i n = [1.17609]^, cos(D); D\ k cos 7) tan ->j = -- ; cos !> = - . n A Corresponding Greenwich mean time = t + [3.55 C30] - sin to have a different sign from D^ : upper under !( immersion ) . sign when an < V is observed. ( emersion j II. Occupation of a Star by the Moon. 6. With the estimated longitude find the corresponding Greenwich and thence take out the moon's horizontal parallax P, and her declination Z/, roughly to the minute ; also, sid. time = apparent time + O's right ascension ; or, sid. time = mean time -f- sid. time mean noon, from p. III. of ephemens + accel. on Greenwich mean time ; h sid. time a, in arc ; P' = ?P', a being the star's right ascension. ' p=P' cosl&mh', Akm min. = [7.92082]-^ ; (h) = h A A; K = P' sin I cos S ; x' P' cos I sin S cos (k) S = d + x x' ; A a in time = [8.82391] ^- ; a = a 8. In the hourly ephemeris of the moon fix on a convenient time (t) at which the moon's right ascension is near to a , and for this time take out TERRESTRIAL LONGITUDE. 221 the right ascension (A), the declination (Z>), and their hourly variations -4,, DI. Then, m = ^SLJ. ; t o = ( t ) + [3.55630] m ; n = [1.17609] A! cos (D) ; cos = [0.56463] ^^. p Corresponding Greenwich mean time = t + [2.99167] sin (r\ =p 4,). Practical Rules for Calculating the Longitude from an Observed Occultation. With the estimated longitude find the corresponding Greenwich time loughly to the minute, and for this time take out from the ephemeris the moon's declination roughly to the minute, her horizontal parallax to the tenth of a second, and the sun's right ascension in time to the nearest sec- ond. To the sun's right ascension add the apparent time of the observa- tion, which will give the right ascension of the meridian. The difference between this right ascension and that of the star will give the hour angle of the star in time, which must be reduced into arc in the usual manner it will be ' > when R. A. of meridian is < f > than R. A. of *. E. ) ( less j Reduce the latitude of the place by subtracting the correction found in the table in Appendix XL, p. 336, for which the nearest correction found in the table will be sufficient. To the proportional logarithm of the moon's horizontal parallax, add the correction answering to the latitude in the following series: Lat. . 11 19 24 29 34 38 42 46 50 54 59 64 69 77 90 Corr. . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 To the proportional logarithm of the horizontal parallax, so corrected, add the log. secant of the reduced latitude and the log. cosecant of the hour angle. To the* sum (,) add the log. cosine of the moon's declination and the constant log. 0.3010. The result will be the prop. log. of an arc, which, subtracted from the hour angle, will give the hour angle coirected. To the corrected prop. log. of the horizontal parallax, add the log. secant 222 SPHERICAL ASTRONOMY. of the *'s leclination, and the log. cosecant of the reduced latitude. To the same log. add the log. cosecant of the *'s declination, the log. secant of the reduced latitude, and the log. secant of the hour angle corrected. These sums will be the prop. logs, of two arcs. The former arc to have the same name as the latitude. The latter to have a different name from the same name as !( less ) the dec. when the h. an^le is \ v than 90 C { greater j The sum of these two arcs, having regard to their names, will give the correction to be applied to the *'s declination to get the declination corrected. To the sum (S { ) add the constant log. 1.1761, and the log. cosine of the * 's declination corrected ; the sum will be the prop. log. of an arc in time, to be added to subtracted from !the *'s R. A., when it is x [ of the meridian, (east ) to get the * 's right ascension corrected. In the hourly ephemeris of the moon, fix on a convenient time at which her right ascension is near to that of the star corrected ; and, for this time, take out the right ascension, the declination, and their hourly variations. Subtract the common log. of the difference between the corrected right ascension of the star and the right ascension of the moon, from the com- mon log. of the hourly motion in right ascension ; to the remainder add the constant log. 0.4771 ; to the same remainder add the prop. log. of the hourly motion in declination. The former sum will be the prop. log. of a time to be added to i . . ( greater ) , ., , > the assumed time when %. s R. A. is hourly var. when % s R. A. is < V than ]) s R. A, a different name from \ } less ) To the common log. of the hourly motion in right ascension, add the log. cosine of the D 's corrected declination ; and to the sum (S 2 ) add the pi-op, log. of the hourly motion in declination and the constant log. 7.1427. TERRESTRIAL LONGITUDE. 223 The result will be the log. cotangent of the first orbital inclination,* and must take the same name as a different name from (hourly motion in dec. when * is ! , I of ( south j To the prop. log. of the difference between the star's declination cor- . rected and the moon's declination corrected, add the constant log. 9.43"54 t and the log. secant of the preceding orbital inclination ; and from the sum deduct the prop. log. of the horizontal parallax. The remainder will be the log. secant of the second orbital inclination,! which must have the name S.1 N. ... , immersion when the observation is an ' emersion. Add together the two orbital inclinations, having proper regard to their names ; and to the log. cosecant of this sum add the preceding sum (/S r 2 ), the prop. log. of the horizontal parallax, and the constant log. 8.1844. The sum will be the prop. log. of a correction to be applied to the time corrected to get the mean time at Greenwich : it must be added ) ( N. , > when the sum of the orbital inclinations is < subtracted ) ( S. By applying the equation of time from p. II. of the ephemeris, there will result the Greenwich apparent time, and the difference between it and the apparent time of observation will show the longitude of the place from Greenwich ; it will be W. E. J t \ i cfreater ) ,, > when the Greenwich time is V f J- than the observed. i. ( j less j Examples. I. SOCAR ECLIPSE. For a solar eclipse, take the example directly calculated in Appendix XL, page 412: Suppose the beginning of the solar eclipse on May 15, 1836, to be observed to take place at i h 36 m 35 -6 p. M., apparent time, in latitude 55 67' 20" N., and longitude about i a m "W. * With the parallel of declination. f With the moon's limb. 224 SPHERICAL ASTROtfOM* Here we have h. m. Observed apparent time . i 36-6 Longitude 12-0 A = +i h 56 m 35 S 6 Greenwich apparent time i 48-6 =+ 24 8' -9 Equation of time ... 3 9 Greenwich mean time . . i 44*7 We hence take from the ephemeris, a = 3 h 29 19", <5 = +i8 67' -t ff = i5' 49"-9, -# = +19 19', P = 54' 24'' -4, 7r = 8".5, P 77 = 54' i5"9. Latitude + 55 5 7 ' 20" Reduction 10 28 . + 55 46 52 . . . . = 9-99902 p 573.51267 cos (h) +9 -96060 . +9-96060 p . 9.99902 cot/ 4-9-83256 cos I + 9- ySooi P' . 3.51169 0-f3i 5o-7 tan 6 -{-9.79316 G +9-71061 cos^ 9-7500T sinA+9-6n83 0+<5+5o 48-3 cos +9-80069 B +9-78899 p +2-87353 (i) +9-92162 check +9 -921 6-J cos D 9-97484 tan (7<)+9- 64936 + 2-89869 tan M +9- 57098 k+ 4 8.0 const. 7-92082 cos M + 9 T^ . +9.97,80 A & + 6-6 +0-81951 tan (0+(5)+o- 08861 cose +9-81719 (h) +24 2-3 +48 58'. 3 tan +0-06041 B +9-78899 *+Iw p> +L!i!5 + 19 3o-9 cos 9 9743o (2) a . i5' 49' '9 + 3 -8 99 23 (i)-(2) A(T 1 1 -6 P . 3-5i38o const. 8 -82391 ) o i' 33". i + 19 3i 34 .0 (t) 3 o o t 3o -f 2 5o i -8 19 3i 3 9 .8 38 9 cos const. 9.97428 i > i 7609 n . o v . . 19 4i 2 tan n . COS rj . k . A . . 9 ;p . . 112 36-4 sin A . const. . 3.t 9 48 7 (2) . ^5363 (i) . + 9-97384 . i -99520 i - 96904 3.26196 . 8-70708 . 9.96528 3-26196 3-55630 6- 7 8354 (3) corr. r 4 ra 38 s - 5 3-5886 7 (3) (2) < -f-corr. + i 45 23-3 Greenwich mean time. 3 56 o Equation of time. i 49 19 -3 Greenwich apparent time, i 36 35 -6 Observed " " Longitude 12 43 > 7 W. of Greenwich. 15 226 SPHERICAL ASTRONOMY. II. OCCTJLTATION OF A STAR. Suppose, at Bedford, on January 7, 1836, in latitude 52 8' 28" N., the immer- sion of i Leonis to be observed at io h 39m 22 S '4 P. M., apparent time, and tho estimated longitude to be about o h i m "W. Required the longitude? 1 Apparent time (observation) Longitude h. m. 10 39 o t W. Latitude Reduc. O N. 62 8 28 10 5 Apparent time (Greenwich Equation of time Mean time (Greenwich) 10 K 5i 5 7 3r Reduced or geocentric latitude. 10 4? For Jan. 7, at io h 4j m , we find, from the Ephemeris, 's R. A. = 19'' I2 m 4o 8 D's dec. = N. i5 5o', and ]> 's equ. hor. par. =56' i"-$. h. m. s. 0's R. A. 19 12 4o P. L. D 's hor. par. o-5o68 App. tim< 3 IO 39 22 corr. for lat. . 9 R. A. meridian .... 5 52 2 P. L. corr' 1 . hor. par . 0-5077 .j C IO 23 26 sec. red. lat. o 2 io3 n s|c's hour iin time . in arc . 4 3i 24 67 5i' cosec. hour angle sum (Si) .... o-o333 cos. J) 's dec. . 9-9832 const, log. o-3oro corr". . . 17 . . P. L. corr". . . . 7^355 sjc's hour angle E. corr d . . . 67 34 P. L. cort d . hor. par. ' o-5o 77 sec. j|c's dec. . O'OiSo 0-5876 cosec. red. lat. oio37 sec. o 2io3 O ' " N. o 42 33-o P. L. 0-6264 sec. corr d . hour angle o-4i84 S. o 3 23-9 P. L 1-7240 corr". . N. o 39 9-1 sum (Si) .... o. 7 5r3 ilc's dec K i4 58 38-8 5Jc's dec. c cos o-o836 iorr''. N. 1 5 37 47*9 corr". . O 1 ' 2m 1 2 s -56 P. L. corr". . y y u --" j 1.9110 #'s R. A. . . 10 23 26 .39 *'s R. A. corr d . 10 2t i3 -83 On referring with the >}c's corrected R. A. to the hourly ephemens of the moon, it will evidently be most convenient to take out the data at n h ; for this time \ve have D 's R. A. lo' 1 2o m 58* -47, hourly motion D 's R. A. 2 2" -9, ]) '* dec. = N. 1 5 47' u"-o, hourly motion D's dec =S. i' 4i"-5. TERRESTRIAL LONGITUDE 227 h. in. s. ijc'scorrd. R. A. 10 21 i3-83 P'sR. A. , . 10 20 58-4? diff. . . o i5*36 ( common log . . i-i864 com, log. h. m. } 's R. A. 2-0896 Remainder . 0.9082 ,.,....,,... ooo32 const.log.. . . . 0.4771 P. L. h. m. 1> 's dec. . . . 1-1874 h. in. 8. o corr". . . o 7 29-9 RL. i-38o3 corr". . S. o i 27-7 P. L. 2-090*: Time assumed 1 1 o o D 's dec. . K". 1 5 47 1 1 o Time corr a . . u 7 29-9 J> 's dec. corr'', K i5 45 43-3 com. log. h. m. D 's R. A. cos. D 's corr d . dec. . 2-0896 . 9'9 83 4 jfc's corr d . dec. . D's " " . O ' . N. 5 3 7 47^9 . N. i5 45 43-3 sum (S) . 2-0730 's hor. par. . o - 5o68 const, log. . . . $.i844 h. m. is. corr". o 19 44-5 P. L. . . , .9599 Time corr d . n 7 29-9 Greenwich mean time 10 47 45-4 Equation of time . . 6 3 1 o Greenwich app. time 10 4i i4'^ Observed * " 10 39 22>4 Longitude . . , t 5a-o "W P. S. The principle of reversing the effect of the relative horizontal parallax on the position of the sun, instead of using the actual effect on the position of the moon, may be advantageously employed in the direct calculation of an eclipse fof a particular place. It will only be necessary to use the parallaxes for the sun viewed as an apparent position, and to diminish the semi-diarneter by the amount derived from the table on page 360. Thu?, it appears, at the beginning of the eclipse, for instance, that the contact may be mathematically tested in two ways. First, we may apply the actual effects of the parallax to the true position of the moon, then augment her semi-diameter, and thus establish ft contact of the limbs. But, if we reverse the operation, and consider the sun to be an apparent body under the influence of the relative parallax^ then clearing it from this supposed 228 SPHERICAL ASTRONOMY. influence by reversing the parallax, and diminishing the semi-diameter, a contact will similarly be established with the true limb of the moon ; and this principle, in its application to solar eclipses, possesses an advantage similar to that derived in the case of an occultation, by considering the star as an apparent place. (See Appendix XL, page 899 )* The formulae, Nos. 2, 3, 4, and 5, pp. 406, 40Y, may, according to thi& method, be supplied by the following : 2. P' = P (P-r); m = P'cosl; $, = [9.4180] ; Q 2 = [9.4180} m sin d ; s= [9.43537] -P. cos JJ A A in minutes = [7.92082] k sin h ; tan 6 = cos (A) cot l\ Gf = cos (h) co t, , tan M= -r. FX tan (A) ; tan s = tan (& + 8) cos M\ cos (6 -f- o) Check . . = cos M cos s ; sind . cos &+o B' (f Q =.(f diminution for s ( partial ) . j s -f phase, Z. = 1 ( total or annular ) ( 5 ^* tf . A a t = ^! A? cos h ; A 5, = Q. 2 sin (A). o = 5 -f- A <5 ; a' = a A a ; y = (a A a) cos'Z> ; 3^ = (a t A a,) cos D ; a: =? (Z> + a' corr.) <5 ; ^ = D l A ^. * This was inadvertently ascribed to Carlini. Professor Henderson, by whom a paper has appeared upon this very point in the Quarterly Journal for 1828, page 411, informs me that, the method has been long in practice, and that it was employed at an early period by Dr, Maskelyne. CALENDAR. 229 734. Longitude ly Eclipses of Jupiter's Satellites. The eclipses of Jupiter's satellites are computed in advance, and the times of occurrence inserted in the Nautical Almanac, to facilitate the determination of terres- trial longitude. After ascertaining, by inspection, about the time an eclipse begins and ends, the satellites are watched with a good telescope, and the precise local time of entrance into and departure from the shadow noted as nearly as possible. The time given in the Almanac, diminished by this observed local time, is the longitude ; west, when the difference is positive, east when negative. This method for finding longitude is defec- tive, for reasons stated in 497. CALENDAR. 735. To divide and measure time and to note the occurrence of events in a way to give a distinct idea of their order of succession and the intervals of time between them, is the purpose of Chronology. 736. All measurements require standard units. These units are, for the most part, purely arbitrary, and are equally convenient in practice. But such is not the case in chronology. Time is divided and marked by phenomena which are beyond our control, and which indeed regulate our wants and occupations. The alternation of day and night forces upon us the solar day as a natural unit of time. 737. To avoid the use of numerous figures in the expression of great magnitudes, all measurements must have their scales of large and small units, and usually the selection of the larger is as arbitrary as the smaller ; Out here the phenomena of nature again interpose, and the periodical return of the seasons, upon which all the more important arrangements and business transactions of life depend, prescribes the tropical year as an- other and higher order of unit in chronology. 738. But the solar day and tropical year are both variable, and are therefore wanting in all the essential qualities of standards. Neither are they commensurable the one with the other ; they are on this account unfit units for the same scale. .Iu the measurement of space, for instance, each unit is constant, and one is an aliquot part of another a yard is equivalent to three feet, a foot to twelve inches, ! .00776 X 400 = 3 d .104. The Pope ordered that the day following, the 4th of October, 1582, should be called the 15th instead of the 5th. This brought the date of the sun's entering the vernal equinox to what it was in 325, the time of holding the Council of Nice. And to secure this coincidence in future, he also ordered that three intercalary days should be omitted every four hundred years, the omissions to take place in those centennial years which are not divisible by 400 ; so that 1700, 1800, and 1 900, which by the Julian mode of reckoning are bissextile, are made b 1 * the Gregorian common years. There is, therefore, at the present time, viz., in the 19th century, a difference of 12 days between the Julian and Gregorian dates. The mode of reckoning by the Julian calendar is called Old, and that by the Gregorian New Style. New style is followed throughout Christendom except in Russia, where the old style is pre- served. 744. Solar Cycle. This is a period of 28 Julian years, after the lapse of which the same days of the week in the Julian system would return to the same days of each month throughout the year. For four such years consist of 1461 days, which is not a multiple of 7, but 7 times 4 or 28 years is a multiple of 7. The place in this cycle for any year of A. D. is found by adding 9 to the year, dividing by 28, and taking the remainder. When there is no remainder, the number sought is 28. 7^5. Lunar Cycle. This is a period of 19 years or 235 lunations which differ from 19 Julian years only by about an hour and a half; so that, supposing the new moon to happen on the first of January in the first year of the lunar cycle, it will happen on that day or within a very short time of its beginning or ending again after the lapse of 19 years. The number of the year of the lunar cycle is called the golden number, to find which tdd 1 to the number of the year A.D., and take the remainder after dividing by 19. If there be no remainder, the golden number will be 19. The golden number is used in ecclesiastical dates to determine the civil date of Easter. 746. Cycle of Indiction. This is a period of 15 years, used in the courts of law and in the fiscal organization of the Roman empire, and thence introduced into legal dates as the golden namber into the ecclesias- tical. To find the place of any year of A. D. in the cycle of indictkm, add 3, divide by 15, and take the remainder. If there be no remainder, the number sought will be 15. 232 SPHERICAL ASTRONOMY. 747. Julian Period. The product of 28, 19, and 15 is 7980. This is called the Julian Period ; and it is obvious that after this period, the years of the solar, lunar, and indlctloi. cvciea will recur in the same order ; that is, each year wil holci the same place in all the three cycles as the corresponding year in the previous period. 748. As no common factor exists in the numbers 28, 19, and 15, it is plain that no two years in the Julian period can agree in its three compo- nent cycles, and to specify the number of a year in each of the latter is to specify the number of the year in the Julian period, which now embraces the entire authentic chronology. The first year of the current Julian period, or that of which the number of the three subordinate periods is 1, was the year B. C. 4713, and noon of the 1st of January of that year, for the me- ridian of Alexandria in Egypt is the chronological epoch to which all his- torical eras are most readily referred, by computing the number of integer days intervening between it and Alexandria noon of the days which serve as the respective epochs of these eras. The meridian of Alexandria is chosen, because it is that to which Ptolemy refers the commencement of the era of Nabonassar, the basis of all his calculations. 749. Given the year of the Julian period, those of the subordinate cycles are found as above. Conversely, given the year of the solar, lunar, and indiction cycles, to determine the year of the Julian period, proceed as follows, viz. : Multiply the number of the year in the solar cycle by 4845, in the lunar by 4200, and in the indiction by 6916, and divide the sum of the products by 7980, and the remainder will be the year of the Julian period sought. 750. A date, whether of a day or year, always expresses, as before re- marked, the day or year current, not elapsed ; and the designation of a year by A. D. or B. C. is to be regarded as the name of that year, and not as a mere number designating the place of the year in a scale of time. Thus, in the date January 5, B. C. 1, January 5th does not mean that 5 days in January have elapsed, but that 4 have elapsed, and the 5th is current. And B. C. 1, indicates that the first day of the year so named (the first, current before Christ) preceded the first day of the common era by one year. The scale A. D. and B. C. is not continuous ; the-year 0, is wanting in both parts, so that supposing the common reckoning correct, our Saviour was born in the year B. C. 1. 751. Epact. The mean age of the moon at the commencement of a year is called the epact. It is a name given to the interval of time be- tween the first of the year and the next preced ng mean new moon: it is expressed in days, hours, minutes, and seconds. Its use is to find the days CALENDAR. 233 of mean new and full moon throughout the year, and thence the dates of certain church festivals. 752. Equinoctial Time. Astronomical time reckons from noon of the current day ; civil, from the preceding midnight. Astronomical and civil dates coincide, therefore, only during the first half of the astronomical and last half of the civil day. Were this the only cause of discrepancy, it might be remedied by shifting the astronomical epoch to coincide with the civil. But there is an inconvenience to which both are liable, inherent in the nature of the day itself, which is a local phenomenon, and commences at different instants of absolute time under different meridians. In conse- quence, all astronomical observations require to be given, to render them comparable with one another, in addition to their date, the longitude of the place of observation from some known meridian. But even this does not meet the whole difficulty, for when it is Monday, 1st of January, of any year, in one part of the world, it will be Sunday, 31st December, of the preceding year, in another part of the world, so long as time is reckoned by local hours. The equivoque can only be avoided by reckoning time from an epoch common to all the earth. Such an epoch is that which marks the passage of an imaginary sun having a mean motion equal to that of the true sun, through a mean vernal equinox receding uniformly upon the ecliptic with a motion equal to the mean motion of the true equinox. Time reckoned from this epoch is called equinoctial time. Equinoctial time is therefore the mean longitude of the sun converted into time at the rate of 360 to the tropical year. APPENDIX. 236 APPENDIX I. cu CO 00 05 < ot tO CO to S o - o CO -H 00 OS CO CO H G^ to ""^ Gx M* ^T t^ CO o 11 I 1 01 o 11 W O - 00 o 00 to OJ O <5 " . ~~ co **3* CO OJ g s 1^ . o to CO 05 CO 3 w 2 - to to to OJ to to S < ^ ? 5 00 OS S OJ CO M ^ 00 CO * OS 05 J^ 00 3 to CO CO e^t Ofc CO to 5J fa g W ^2 Q O - CO r 1 5 CO 10 CD o CO oo ^ H S CO co CO 00 1 CO to CO 1-1 1-1 (.M 1-1 CO __ H fc CO 00 o g 2 fif OJ to CO OJ OJ CO o o g - " w ol * 00 CO 2 fc 3 S w ^ S OS S? p 00 CO * g 00 CO OJ 00 to Q to OJ CO OJ < } OS g tfd CO oo CO CO CO CO CO CO 00 a g o o o vr. o o CD o o .0 o o o o o o .i OS CO OS CO r | o CO o CO OQ Tf S 1 CO 1 S 1 -t o 22 1-^ I ( to OS 05 o CO o S 03 1 OJ o o o t- o oo o g o o o o 2 CO o 00 oo ^ o CO g g o * -" ^ OS 00 Tf CO m OS on CO 00 OS to CO * OS r qgl CO 10 OS ~~ l OJ to S- ' ^ r- CO oo CO 00 oo 00 S $ 2 to CO 1 1 1 1 fc ri >^ . %t 0) N w h 5 ss o 5 I 00 ^3 & 'a, 1 a a OJ 3 cL 3 j^ K^ w ^ 3 r^ j-J J? P< C5 I Of *o s * * & APPENDIX II. 237 APPEI^ DIX II. ASTRONOMICAL INSTRUMENTS. Astronomical Clock and Chronometer, 1. The order and succession of celestial phenomena make time a most important element in astronomy, and accordingly the utmost scientific and mechanical skill has been devoted to the perfection of instruments to indicate and measure its lapse ; 37. The best time-keepers now in use are the Clock and Chronometer. Both consist essentially of a motor, a combination of wheel-work to transmit and qualify the motion it impresses, and a check, alternately to arrest and liberate the movement, and thus to mark an interval designed to be some aliquot part of a day, the natural unit of duration. 2. The Clock. --Iu the clock, the motor is a weight A suspended from a cord wound about the drum B of a wheel (7, and the check is the anchor escapement JV, controlled by the vibrations of a pendulum P, whose rod is geared to an arm projecting from the axis 0, with which the anchor is firmly connected. The weight A turns the drum B and its wheel (7; the wheel C turns the pinion D and its wheel E\ the latter turns the pinion F and its wheel 6?, and so on to the pinion L and its wheel Mj called the scape-wheel, of which the teeth are considerably under- cut, so as to turn their points in the direction of the motion. The flukes of the anchor are turned inward, forming two projections called pallets. The distance between the ends of the pallets is less than that between the points of two teeth that lie nearest the line drawn from one pallet to the other ; and no wo teeth can, therefore, pass the same pallet without the wheel being arrested by the contact of a tooth op the opposite side with the other. With the swing of the pendulum the anchor oscillates, and one pallet is thus made to approach while the other recedes from the wheel. As soon as the receding pallet disengages itself from a tooth, the wheel is turned 238 SPHERICAL ASTIION'OMY Fig. 6. by the motor and intermediate machinery till arrested by the approaching pallet, now interposed between its teeth on the opposite side. The re- turning swing of the pendulum reverses the pallet motion, liberates the wheel long enough for another tooth to pass, and again arrests it, and so on. Thus, by regulating the length of the pendulum and number of teeth on the scape-wheel, an index or hand connected with the arbor of the latter may be made to travel by successive leaps, as it were, around the <'ircu inference of a circle on the dial-plate in any given time, 3, If the anchor be connected with the seconds pendulum, and there be sixty teeth on the wheel, each leap will mark a second. The APPENDIX II. 239 motions of the minute and hour hands are regulated by suitably propor- tioning the relative dimensions of the intermediate wheels with whose arbors these hands are connected. 4. The scape-wheel being in a state of constant tension by the incessant action of the motor, its teeth must act upon the pallets first by a blow and then by a pressure during the time of contact. The bearing surfaces, of which there are two on each pallet, inclined to one another, are so cut that the direction of the blow on the first from the tooth of the scape-wheel passes through the axis of the pendulum's motion, while the pressure from the same tooth on the second passes clear of that axis and accelerates the motion in the direction of the swing, thus restoring whatever of loss may come from friction and atmospheric resistance. 5. The pendulum bob possesses the principle of compensation. It consists of a cylindrical glass vessel resting upon a plate at the end of the pendulum rod. This vessel is filled with mercury to a depth so adjusted to the length of the rod as to elevate by its expansion or depress by its contraction the centre of oscillation just as much as this centre is depressed by the expansion or elevated by the contraction of the rod during a change of temperature. The distance between the axes of suspension and of oscillation being thus made invariable, the time of vibration will con- tinue constant, and the check be interposed at equal intervals. 6. Chronometer. The chronometer is an accurately constructed balance watch, uniting great portability with extreme accuracy. It is of various sizes, the larger having dial-plates from three to four inches in diameter, and running from two to eight days between the windings. The larger kind are suspended upon gimbals to secure uniformity of position, are mounted in boxes, and are called box chronometers. The smaller kind resemble in shape and size a common watch, are worn in the pocket, and are called pocket chronometers. 1. The motor is an elastic spiral spring inclosed in a short cylin- drical box A, called the barrel, one end being permanently fastened to a stationary axis E, about which the barrel freely turns, and the other to the inner surface of the barrel. The barrel being turned in the direction of the coils of the spring, the elastic force of Ae latter is brought more and more into play, and its variable action thus produced is communicated by means of a chain B tc a variable lever (7, called a, fusee, whose office is to modify and transmit il uniformly to the works of the instrument. The fusee is a conical solid having its surface broken into a spiral shoulder, running from one end to the other, the curve being so regulated SPHERICAL ASTRONOMY that the distance of any one of its points from the axis of the fusee's mo- tion multiplied into the force of the spring, acting through the intermedium of the chain, shall be a constant quantity ; and as the main wheel D, which gives motion to the rest, is firmly secured to the fusee, the motion is made to act uniformly upon the instrument. 8. The swings of the pendulum by which the check was alternately interposed between and withdrawn from the teeth of the scape-wheel in the clock, are, in the chronometer, replaced by the vibrations of what is called the balance. This consists of a wheel , freely movable about an axis (7, and a thin spiral spring S, one end of which is securely fastened to the hub of the wheel, and the other to a fixed support A. If when the spring is free from tension, the wheel Fig. 8. be brought to rest it will remain so, just as a pendulum bob brought to rest at its lowest point will remain im- movable. If from this position of the wheel it be turned in either direction about its axis, the spring will wind or unwind, the elastic force of the spring will be called into play, and will, when the wheel is unobstructed, carry it back to its position of equilibrium. But having reached this position, its living force carries it beyond ; the action of the spring is reversed, and, after destroying the living force, will reverse the motion; the wheel will return to its position of equilibrium, which it will reach with a living force equal to that it had befo-e at the same place, but in a contrary direction. The wheel will*pass on, the action of the spring be reversed, the wheel will return as before, and thus the vibrations be continued forever, as in the case of the pendulum, but for the waste of living force from friction, atmospheric resistance, and absence of perfect elasticity in the spring. 9. The angular acceleration impressed upon the balance by the spring APPENI/IX II. 241 is measured by the moment of its elastic force divided by the moment of inertia of the entire balance. When the temperature is increased, the spring is lengthened and the elastic force it exerts lessened ; the wheel is expanded, its matter thrown further from the axis of motion, and the mo- ment of inertia consequently increased. On both accounts the angular acceleration is diminished, and the balance will vibrate slower, and the intervals 'between the checks be increased. The effect is just reversed when tho temperature is diminished. This is the source of greatest difficulty with all portable time-keepers, and renders the common watch worthless for any thing beyond an approximate indicator of the time. 10. To remedy this defect, the common wheel is replaced by what is called an expansion balance, which is re- Fi gi 9 - presented in the figure. A A is a bar which receives the end of the arbor into an aperture at its middle point. To the ends of the bar are securely attached two compound metallic curves C C, composed of two concentric strips, one of steel and the other of brass, the latter being on the convex side ; these are soldered or burned together throughout their entire length. Each of these curved pieces carries a heavy mass D I), movable from one end to the other, but capable of being secured in any one place by means of a small clamp-screw shown in the drawing. Now when the temperature increases, the exterior brass expanding more than the interior steel, the ends C C are thrown inward towards the arbor, while the ends of the bar are thrown outward, but through a much less distance ; and thus by properly adjusting the places of the masses D D, the moment of inertia of the balance may be made to vary directly as the moment of the elastic force of the spring ; in which case the angular ac- celeration becomes constant, and the intervals between the interposition of the checks equal. 11. To regulate the rate, two large-headed screws B B, calk*! mean- time screws, are inserted, one into each end of the bar. If the chronometer run too slow, the moment of inertia is too great for that of the spring, and these screws must be screwed up, which has the effect to lessen the dis- tance of their heads from the axis of motion, and thus to lessen the mo- ment of inertia, and increase the angular acceleration. If the chronometer run too fast, the screws must be unscrewed, the effect of which must be obvious. 16 SPHERICAL ASTRONOMY. 12. The escapement is of the kind usually called the detc ched, from the fact that except at certain instants of time, the whole appendage of the balance-spring is relieved from the action of the scape-wheel. The scape-wheel is represented at'Jkf ; it is urged by the motor, acting through the wheel-work, to move in the direction of the arrow-head. A steel roller (7, called the main pallet, is firmly fixed to the arbor of the bal- ance. In the pallet is a notch i, having one of its faces considerably un- dercut, and . covered with an agate or ruby plate to receive the action of the teeth of the scape-wheel. Securely fixed to one of the frame-plates of the chronometer is a stud B, and to this is attached a spring A, called the detent ; this spring is extremely thin and weak at the stud B. Attached to the detent is a stud D. A ruby pin projects from the detent at c, which receives a tooth of the scape-wheel when one escapes from the pallet bear- ing i. From the stud D proceeds a very delicate spring E, called the lifting spring, which rests upon and extends beyond a projection F from the end of the detent ; this projection being so made that the lifting spring cannot move in the direction from the scape- wheel without taking the detent with it, and thus lifting, as it were, the pin c from the tooth with which it is in contact, while it leaves the lifting spring free to move towards the scape- wheel without disturbing the detent. Concentric with the main pallet, at- tached to and just above it, is a small projecting stud a, called the lifting pallet, which is flattened on the face turned from the scape-wheel and rounded on the other. The flattened is called the lifting face. 13. Mode of Action. In the position of the figure, the main pallet, under the action of the balance-spring, is moving in the direction of the arrow-head/, and the lifting pallet is coming .with its lifting face in contact with the lifting spring E, which it lifts with the detent so as to raise the pin c clear of the tooth of the scape-wheel with which it i? in contact, APPENDIX II. 243 By the time the wheel is free from the pin c, the main pallet has advanced far enough to receive an impulse from the "tooth .t upon its jewelled surface. ij and before this tooth escapes, the lifting pallet a pails with the lifting spring E, and the detent returns to its place of rest and interposes the pin c to receive the tooth t ( as soon as the tooth t has been liberated by the onward movement of the main pallet from its face i. The balance having performed a vibration ty the impulse given to the main pallet, returns by the action of the balance-spring, and with it the lifting pallet 0, whose rounded face, pressing against the lifting spring E, raises it and passes, first the detent without disturbing the latter, then the lifting spring, and moves on till the balance has completed the vibration, when it returns to the po- sition indicated in the figure, and the same evolution is performed again ; the balance thus making two vibrations for every impulse. The Vernier. 1. This is a device by which the value of any portion of the lineal distance between two divisions of a graduated scale of equal parts may be found in terms of the space itself. It consists of a scale whose length is equal to any assumed number oj parts of that to be subdivided, and is divided into equal parts of which the number is one greater or one less than the number of the primary scale taken for the length of the vernier. A ^-T-I t ? I--HPB 11 I If Let AD be any scale of equal parts, and denote by s the length of n 1 of these parts ; then will n 1 be the value of the unit of the scale. Take a vernier B E of equal length s, and suppose it divided into n equal parts, then will be the length of one of its parts, and the difference of length between m parts of the scale and an equal number of parts of the vernier, will be ms ms m .s n^l ~ V = n -!' 244 SPHERICAL ASTRONOMY. But is the value of the unit of the scale, and n the whole number of n 1 divisions of the vernier ; denoting the first by F, this difference may be written - F. n ISTow, the length of a part on the scale is greater than that on the vernier, and the number of parts on the vernier is greater by one than the number in an equal length on the scale ; hence r if the m th intermediate division of the vernier coincide with any one division on the scale, the zero of the ver- nier will fall between two divisions of the scale, and be in advance of that bearing the smaller figure by the distance expressed above ; so that, taking the zero of the vernier as the index or pointer, its distance from the zero of the scale will be the number of units denoted by the figure on the division next preceding, plus the th part of the unit of the scale. Thus, in the figure, A being the zero of the scale, B that of the vernier and therefore the pointer, the distance of the latter from the former will be Aa-{-aB', and because n10, and the division b of the scale coincides with the 4th 4 of the vernier, m=4, and the distance AB=Aa-] .fe. 2. The least value that may be read with certainty is obtained by making m=\, which will give, F n' Whence we have this rule for finding the lowest reading by means of the vernier, viz. : Divide the lowest count, or unit of the scale, by the number of divisions on the vernier. If the scale be tenths of inches, and we make w=10, then will 7=^- 10 =i ; in which case the subdivisions will be carried to hundredths of inches. 3. The vernier is equally applicable to all kinds of scales, to circular as well as rectilinear : the only condition being that the different parts shall be equal. Suppose each degree on the circumference of a circle is divided into 6 ojual parts, and that tbo. number of parts on the vernier is 60, then will APPENDIX II. 245 V=10' and n 60 So that the read' ng of angles with an instrument having such a circle may be carried to ten seconds. Micrometer. 1. The Micrometer is an instrument employed to make minute measurements, and is applicable alike to time and linear distance. It has various forms. 1*. The Reticle. He who views a distant object through a telescope, does not look at the object but at its image within the tube of the instru- ment. The image of a point is always in a plane through the focus of the lens conjugate to the point itself, and perpendicular to the tube of the tel- escope. The visible portion of this plane is called the field of view. Some point in the field of view is arbitrarily assumed as an origin of reference, and marked by the intersection of a pair of cross wires. The line through this point and the optical centre of the field lens, is called the line of collimation. 2. If the telescope be at rest and an object in motion, the image of any one of its points will when visible pass across the field of view ; and one of the opaque wires being made to coincide with its path, the image will move directly towards the line of eollimation, and ftie exact instant of its reaching it may be noted. But every such observation is liable to error. To increase the chances of avoiding this error, the wires marking the line of collimation are made perpendicular to one another, and an equal number of equidistant and parallel wires added on either side of that which is per- pendicular to the path of the image. When the motion of the image is uniform, an average of the times of passing the parallel wires will, accord- ing to the doctrine of chances, give a time of passing the line of collima- tion more free from error than the single observation. 3. This simple form of the micrometer is call- ed a reticle. The wires or spider lines are stretched across a circular metallic diaphragm pierced by a large concentric opening. On the edge of the diaphragm, and in the prolongation of the single wire, two studs project at right angles to its plane ; and these, with two antagonistic screws A B, hold the reticle in po- 246 SPHERICAL ASTRONOMY. sition ; the screws, for this purpose, passing through the tube of the teles cope and leaving the heads exposed for purposes of adjustment. 4. Position Filar Micrometer. The purpose of this instrument is to measure the angles at the observer, subtended by the distances between objects that appear very close together, and to determine the positions of the planes of these angles. It consists of two parts, viz. : One to measure the angle between the objects ; and the other, the inclination of the plane :>f the objects and observer to some co-ordinate plane. 5. The first is represented in the figure, a and c are two fine par- allel wires, which are made to move at right angles to their lengths by means of screws firmly connected with the forks A and (?, to whose prongs they are attached. The screws have fifty threads to the inch, and are Fig. 12. moved by nuts sc mounted as to admit of a motion of rotation without translation, so that by turning the nuts a motion of translation is commu- nicated to the wires in either direction, depending upon the direction of the rotation. Thl outer surfaces of the nuts are cylindrical, and enter fric- tion tight the central perforations of two circular wheels whose planes aro perpendicular to the lengths of the screws, and which are large enough to admit of their circumferences being divided into 100 equal parts, which parts are marked and numbered. Each wheel is provided with a station- ary pointer or index. A third and stationary wire, perpendicular to the first two, is supported by a diaphragm disconnected from the forks. Upon one of the interior edges of this diaphragm, and parallel to its wire, is a graduated scale in the shape of a comb, having 50 teeth to the inch, so that one revolution of a nut will carry its movable wire from the centre of one valley between the teeth to that of the next. Near the central valley of the scale is a small hole to mark the zero of the comb-scale, from which the scale is estimated in either direction. It is easily seen that a turn of the nut-head through one of its divisions will move its wire through a linear distance eqiuJ to y-j^ of s*o or 5^ oW f an m h an d having ascertained by the measurement of some small distance on the circumference of a great circle of the celestial sphere, or by the process in Example, p. 249, its equivalent in arc, thi?. the APPENDIX II. 247 micrometer part of the arrangement, is readily applied to the dete 'mi nation of small angles. 6. The second and position part consists of a circular plate A A, called the position circle, some three or four inches in diameter, having its circumference divided into 360, which are again subdivided to any convenient extent. The central part is cut away, and the micrometer arrangement so attached, with its wL es parallel to the position circle, as to admit of a free motion Fig. ia of rotation about an axis through its centre, and per- pendicular to the plane of the wires. To the revolving plate of the micrometer part are attached two verniers V V, and motion is com- municated to the latter by a ratchet and pinion, of which latter the head is seen at 0. The microscope by which the wires and comb-scale are magnified, and which serves also for the eye-glass of the telescope, is represented at E. By means of a screw cut upon a projecting ring .around the large and central aperture of the position circle, the instrument, as represented in the figure, is attached to the tail end of the telescope. 7. To measure the angular distance between two objects in the field of view, turn the head till the fixed wire passes through their images, then bisect the images by the movable wires ; note the reading on the comb-scale and upon the heads; take their sum or difference according as the wires are on opposite sides, or same side of the zero of the comb-scale. This reduced to arc will be the measure sought. Note also the reading of the position circle ; this will give the inclination of the plane of the angle to the plane through the zero of the position circle. A second angle being measured in the same way, the difference between the second and first reading of the position circle will give the inclination of the planes of the two angles. Micrometer Revolution. THE micrometer being supposed ir place, and the eye-piece pressed for- ward far enough to obtain a distinct view of the wires, the telescope is directed to some distant object, and adjusted to distinct vision. An image of the object will be formed on the plane of the wires, and any one of its 24:8 SPHERICAL ASTRONOMY. linear dimensions may be measured by turning the position circle till the stationary wire coincides with, and the movable wires pass through the extremities of its image. The number of entire comb-teeth between the movable wires, multiplied by 100, and this product increased by the sum of the readings of the screw-heads, will give the linear dimensions of the mage expressed in units of the screw-head. The value of the latter is, in /he case we have taken, g-^-Q" ^ an mcn - To find the angle subtended by the object, we must know the angular value of the unit on the screw- head. It is demonstrated (Optics, 60) that the optical image of any point of an object, is on a right line drawn through the point and the optical cen- tre of the lens by which the image is formed. The angles, at the optical centre, subtended by an object and its image, are therefore equal, and if the images of objects which subtend equal angles were at the same dis- tance from the optical centre, they would be of the same size. The lineai dimensions of the images at the same distance from the optical centre, would therefore be proportional to the angles subtended by their respective objects, and to find the angular value in question, it would be sufficient to cause the image of some well-defined object, whose distance and dimen- sions are known, to be embraced by the wires, and to divide the angle which the object subtends, expressed in seconds, as determined trig&nometrically, by the number of units of the screw-heads, which indicate their separation. But the distances and therefore the dimensions of images, whose objects subtend the same angle, are variable, being dependent on the distance of the objects, and from the value found by the above process must be de- duced that which would have resulted had the image been formed at some constant distance, which is that of the principal focus. Let / and /" denote the distances respectively of the object and its image from the optical centre, and F u the principal focal distance of the object- glass, supposed convex. Then, Optics, 44, Eq. (40), fa- ~ ' /"' J and denoting by n and JV, the number of units of the screw-heads when the image is embraced at the distances f and F n respectively, we shall have, Optics, 64, Eq. (58), /" : F u : : n : JIT; whenca F fF *=*..,,-_=_/__ M APPENDIX II. 249 and calling a, the number of seconds in the angle subtended by the object^ we have, by the rule just given. JSxample. The length of the object measured in a direction perpen- dicular to the line of sight was 3 feet ; the distance from the object-glass, 261.9 yards ; the principal focal length, 45.75 inches; and the sum of the divisions on the screw-heads indicating the separation of the wires, 1819. Then /= 261.9 yd8 -; F n = 45.75 in - = 1.2708 yds -; n= 1819. fF n 260.6292 yd3 -. R 5 yds- tan -i a = ' ' -g-, of which the log. is 7.280835 ; whence a= 13' 07".57 = 787".57. Log. a .... 2.8962892 " / . . . . 2.4181355 ' n a comp. . . 4.7401673 " F u " . . 3.5839923 -1.6385843 Now, to measure the angle subtended by the distance between any twe points, direct the telescope so as to get the images of the points in the field, and turn the micrometer till the stationary wire apparently passes through them, and by a motion of the screw-heads bring the movable wires to the images the number of units of '-he screw-head, which indi- cate the separation of the wires, multiplied by the decimal 0".4351, will give the number of seconds in the angle. The value of , being a function of F it ^ Eq. (a), will of course vary with the object-glass, but is perfectly independent of the eye-glass. If the distance/ be so great that F n may be neglected in comparison, then will Eq. (a) give N n, which will be the case when the angular value is determined from astro- nomical objects. 250 SPHERICAL ASTRONOMY. Spirit-Level. 1. This is an instrument used to adjust a line to a given position in reference to the horizon. It consists of a cylindrical glass tube A A, whose axis is the arc of a circle. This tube is filled nearly full with some one of the more perfect fluids, such as alcohol or naphthalic ether, leaving a small portion of air, seen at B, called the air-bubble, and hermetically sealed at both ends. It Fig. 14. is then usually set in a metallic tube (7, very much cut away on one side from the middle towards the ends, so as to exhibit the bubble and fluid when in a horizontal position. This metallic tube is connected with a plate of metal F F, by a hinge E and screw Z>, the axis of the hinge being perpendicular, and that of the screw parallel to the plane of the circular axis of the level. 2. A scale of equal parts is cut either upon the upper surface of the glass tube or upon a slip of ivory and metal lying in the plane of the tube's curve, as represented at G G. The divisions of the scale being numbered, the value of the spaces in arc is readily ascertained by attaching the level to the face of a vertical graduated circle, and turning the latter sufficiently to cause the air-bubble to pass from one end of the scale to the other.' The angular space passed over by the circle reduced to seconds, divided by the number of units on the scale traversed by the bubble, will give the value of the unit in some multiple of the second. 3. Use. The surface of the fluid being always horizontal, the line connecting the ends of the bubble will be a level chord of the level's arc, and the radius passing through the point of the scale midway between the ends of the bubble will be vertical. Now, suppose any line of an instrument with which the level is used to be made parallel either to the radius passing through the zero of the scale, or to the chord whose ends are marked by the same numbers ; then, to make this line vertical in the first case, or horizontal in the second, move the instrument, the level being securely attached, till the ends of the bubble are equally distant from the zero. If the ends of the bubble be not at the same distance from the zero, the inclinat'on x of the line in question to the vertical cr horizontal APPENDIX II. 251 direction is thus found : Let a denote the semi-length of the bubble, m and n the numbers of the scale at its extremities, then will whence m n This value of x being independent of the length of the bubble, which is indeed a variable quantity, even in the same level, because of its varying temperature, gives the inclination of the line under consideration to its proper position, when the level is adjusted to the instrument. If the lower surface of the plate F F be parallel to the chords of equal numbers, the inclination of any given line or plane may be ascertained by laying this plate upon it and applying the above rule. But if the lower surface of the plate be not parallel to the chords of equal numbers, its inclination to them, and that of the plane or line in question to the horizontal or vertical direction may nevertheless be found thus : Denoting the first by y, and the latter, as before, by .r, and using the notation of equation (2), we have for one position of the level, and for the reversed position of the plate with its level, whence _ l+l' _ mn+m'n' I I' m nm'n' If the given surface or line be provided with adjusting screws, as is the case in all astronomical instruments, the ends of the bubble may be brought to the same reading in the first position of the level, in which case, we have m=n, and m' n' * = ~=-y ....... (3) The angle y is called the error of the level, and the angle x the error in level of the instrument, and the above equation gives this rule for and correcting these errors, viz. : 252 SPHERICAL ASTRONOMY. The level beinv placed over the given line, bring, by means of the adjusting screws of the instrument, the bubble to read the same at both ends ; then reverse the level, or turn it end for end, and take one fourth of the difference of the new readings ; add this to the lesser of the read- ings, and turn the screw D till the end of the bubble nearest the zero reach the numb* r answering to this sum, to which add again the same quantity, and bring the end of the bubble to this new reading by the adjusting screws of the instrument. The ends of the bubble will stand at the same numbers, and both errors will be destroyed. Reading Microscope. 1. This instrument, like the vernier, has for its object to read and subdivide the space between two consecutive divisions of any scale of equal parts, and is the most perfect yet devised for this purpose. It is a compound microscope, whose object-glass forms an enlarged image of the space to be divided. This image is thrown upon the plane of two spider-lines or wires, arranged in the form of a St. Andrew's cross, and so placed that a line bisecting its smaller angles is parallel to the cuts or division marks of the scale. The cross is attached to a diaphragm, which is moved by a micrometer screw in the direction of its plane, per- pendicular to the axis of the microscope. The head of the screw is divided into any number of equal parts, depending upon the nature of the scale and the extent to which the subdivisions are to be carried. The numbers on the head are so placed that when the screw is turned in the direction to bring them in the order of their increase to a fixed pointer, the cross shall move along the image-scale in the direction in which its numbers decrease. Within the barrel of the microscope is a stationary comb-scale, like that in the position micrometer. Its plane is parallel to that of the cross, and the distance between the centres of two valleys, separated by a single tooth, is equal to the space over which the cross is moved by a single revolution of the screw. Every fifth valley is cut deeper than the others to facilitate the reading ; and near the bottom of the central valley of the comb is a small circular aperture, to mark the zero position of the pointer or index, which is a small wire attached to the movable diaphragm, and so placed that its prolongation shall bisect the smaller angles of the cross. In (1), A A is the main tube of the microscope, passing through a collar or support B, where it is firmly held by two milled nuts g g, which act upon a screw cut upon the outer surface of the tube. These nuts also serve to change the distance of the whole microscope from the scale to APPENDIX II. 253 be read ; h is the object-glass placed in a smaller tube, upon whose outer surface is also a screw, by which this glass may be moved independently Fig. 15. Fig. 16. 2. of the main tube ; the diaphragm of the cross is in a working box, whose edge is seen at a ; e is the graduated head, firmly attached by a friction clamp to the nut b of the micrometer screw ; / is a pointer attached to the working box ; d is the eye-glass, which moves freely in the direction of the axis of the microscope by a sliding tube ; at c' is represented the head of a small screw, which supports and gives motion to the comb-scale within the working box, and S S represents the edge of the scale to be subdivided. In (2) is represented the field of view, as seen when the eye is applied at e?, in which m m' is the image of the scale, with one of its cuts bisecting the smaller angles of the cross, and e the wire index at its zero position, as indicated by its being seen through the centre of the circular aperture of the comb. In this position of the pointer, the zero of the graduated head e is brought to the index /, by holding the nut b firmly in the hand, and turning the head, which is only held in its place, as before stated, by the action of the friction nut. 2. The quotient arising from dividing the length of the image space by that over which the wires move in one revolution of the screw-head, as given by the comb-scale and head, is called the run of the micrometer. For convenience, the run should be an entire number. 3. The image-scale must be accurately in the plane of the wires, otherwise there would be a parallactic motion, which would shift the position of the wires on the image-scale at every change in the position of the eye, and thus vitiate the measurement. This parallactic motion is easily detected by slightly shifting the position of the eye when looking through the eye-glass. There are, then, two adjustments for the reading microscope, viz., that for the run and that for parallax 254 SPHERICAL ASTRONOMY. 4. The size of the image of an object, and its distance from the lens by which it is formed, are dependent upon the distance of the object from the lens, being greater in proportion as this distance is less, and less HS it is greater. If the distance of the object-glass of the microscope from the scale be changed by means of the screw on the tube at A, the size of the image space will be altered, and may, therefore, be made of such dimensions that the cross will move from one division to the next in order, by a given entire number of revolutions ; and if by this operation, the image be thrown off the plane of the wires, as it in general will, it is restored by changing the distance of the whole body of the microscope from the scale by means of the milled nuts g g. By two or three efforts cautiously conducted, the adjustments may be made without difficulty. To illustrate, let the scale be that of the sexagesimal division of the circle, and suppose each degree divided into twelve equal parts, each space will be equal to five minutes ; if we make the run five, each tooth on the comb will be equal to one minute, and if the screw-head be divided into sixty equal parts, each of its spaces will be equal to one second; so that the circle may be read to seconds. Now suppose on examining the run, which is done by turning the screw-head till the cross moves from one division to the next in order, it be found 5' 10" ; it is too great. Move the object-glass h from the plane of the circle by screwing in its tube, the image will decrease, and, if it were before on the plane of the wires, it will now pass to some position between that plane and the object-glass h. Move the whole body of the microscope by means of the milled nuts gg towards the circle ; the image will be restored to its proper position, with less dimensions than it had before. By one or two repetitions of this process the adjustments are made. 5. The wire pointer at its zero position on the comb-scale is the index of the circle or instrument scale. When the pointer, in this po- sition, is immediately opposite a division mark of the circle scale, say the third after that marked 27, which is indicated by the angles of the cross being bisected by the image of that division mark, the reading is 27 15' 00" ; but if the intersection of the cross wires falls between the third and fourth divisions after that marked 27, then will the reading be greater than that above by the value of the distance from the cross wires to the division mark to which tho cross will move by turning the screw- head in the order of its increasing numbers. To find this value, turn (4ie screw-head in the direction just indicated till 'he angles of ibe cms? ar* APPENDIX II. 255 bisected by the division mark in question, and count the entire number of comb teeth between the aperture and pointer, then note the reading on the screw-head; suppose the former to be 3 and the latter 41, the true reading will be2718'41". The Transit. 1. The transit is an instrument which is used in connection with a time-piece to ascertain the precise instant of a body's passing the me- 256 SPHERICAL ASTRONOMY. ridian of a place. ' It consists of a telescope T T, usually of considerable power, permanently fixed to a substantial axis A A, at right angles to its length. The axis terminates at each end in a steel pivot, accurately turned with a diamond point, to a cylindrical shape. The pivots are of equal diameters, received into notches cut in two blocks of metal, called Ys, which rest in metallic boxes, the latter being imbedded in metallic or stone piers, according as the instrument is intended to be portable or fixed. 2. Permanently attached to the tail or eye end of the telescope, on opposite sides, are two small graduated circles, called finders. The planes of these circles are perpendicular to the axis of the transit, and each circle has an index-arm, which carries a small spirit-level and two verniers, one at each end. The index-arms are movable about the centres of their respective circles, and are, as well as the axis of the transit, provided with a clamping and tangent screw arrangement, thus affording, with the aid of the level and verniers, the means of giving the telescope any de- sired inclination to the horizon. 3. At the solar focus of the object-glass of the telescope is a reticle,Y\g. 11, in which the single is replaced by a double wire, with small interval, and so placed as to be parallel to the axis of the transit. These are called axis wires. Those wires of the reticle which are at right angles to these are called the normal ivires. To the fixed wires of the reticle a movable one is added ; it is always parallel to the normal wires, indeed, is itself a normal wire, and is put in motion in the direction of the axis wires by means of a micrometer screw, with graduated head, shown at m. 4. The small tube containing the eye-piece of the telescope is attached to a slid ing-frame, connected with a screw e, by which the eye- piece is carried from one side of the field of view to the other, in the direction of the axial wires. 5. The axis is hollow throughout, and the pivots are perforated at the ends to admit the light from a lamp L, supported upon one of the piers. This light is received by a reflector within the tube of the tel- escope, and inclined to its axis under an angle of 45, and is reflected to the eye-glass, thus illuminating the field of view, and exhibiting the wires of the reticle. The reflector is perforated by an elliptical opening in its centre, to permit the direct light from any external object to pass freely to the eye end of the telescope. When the illumination is through the other end of the axis, the reflector is revolved through an angle of 90, by means of a milled-headed wire, with which it is permanently con- nected. The head is shown at r. APPENDIX II. 257 Fig. 18. Fig. 19. Fig. 20. G. The boxes which support the Ys are large enough to permii i slight play in the latter; one in a horizontal, Fig. 18, and the other in a vertical direction, Fig. 19, the motions being effected by antagonistic screws. By the first of these motions, the line of collimation is brought to the meridian, after the rougher ap- proximations to that plane are made by other means, and by the second the axis is made horizontal by the aid of a large and delicate spirit- level, Fig. 20, mounted upon in- verted Ys, far enough apart to rest upon the pivots. Adjustments. \ 7. The transit is adjusted within itself when its line of collimation is perpendicular to its axis ; and it is in position, when its axis is perpen- dicular to the meridian. Its finders are adjusted, if the air-bubbles at their levels indicate the same reading at both ends, when the verniera indicate the true inclination of the line of collimation to the vertical or horizon. 8. It is by no means necessary, or even desirable, to aim at perfect adjustment. It will, in general, be much safer to- reduce the errors of adjustment to narrow limits, then to determine their amount, and eliminate their effect from observation, in the manner to be described presently. 9. Line of Collimation. Direct the telescope to some small, distant, and well-defined terrestrial object. Bring it apparently between the horizontal wires, and measure its distance from the central normal wire by means of the micrometer and movable wire; denote this dis- 17 258 SPHERICAL ASTRONOMY. tance by c'. Lift the transit from its Ys, turn the axis end for end, and measure, as before, the apparent distance of the same object from the middle wire, and denote this distance by c". Place the movable wire at the distance, of c'+c" on the side of the object from the middle wire, and move the whole reticle by the antagonistic adjusting screws, which lie in the direction of the axial wires, till the object appears on the movable wire ; the line of collimation will be adjusted. 10. Error of this adjustment. If n denote the value in arc of the micrometer's unit, then will the angle which the line of collimation makes with its proper position, before moving the diaphragm, be (4) and the line of collimation will describe, when the telescope is moved, a conical surface, whose intersection with the celestial sphere will be a small circle. Example. When the telescope is pointing to the south, let the middle wire appear to be 326.3 revolutions to the right hand of the object ; when the axis is reversed, let it appear 318.7 to the right, then will 326.3-318.7 n . = c= and if one revolution of the micrometer correspond to the space an equa- torial star would pass over in three seconds of time, then will 3s.Xl5 = - T _ = 0'.45 ) and c=3.8xO".45 = l".7l. 11. The axis. This must first be levelled, then moved in azimuth till it is perpendicular to the meridian. Mount the level with its inverted Ys upon the pivots, bring the bubble to the same reading at each end by the adjusting screw of the level ; reverse the level, and bring the bubble again to the same readings half by the screw of the level and half by the vertical antagonist screws of the Y, which admits of vertical motion. Repeat the operation once or twice, and the thing is done APPENDIX II. 259 12. The error in ills adjustment. After the first approximation, denote by e', e" , &c., the reading of the east end of the level; by w', w", . ' * - ~~, - > . -- r - , sin Ps' cos 5 representing the zenith distance by (X ), to which it is nearly equal, and regarding Vs' as the altitude, from which it differs but by a very small quantity. The angle s"P is given by equation (9), Z denoting as before the azimuth error. Whence, denoting by ~t the time of observation, we obtain for the time of meridian passage __o_ J_ cosjX-5) Z sinjX-5) "^15. cos ^"^ 15 ' cos S 15* cos 6 in which c, , and Z may be found in the manner already indicated, or still better as follows. Making (16) and supposing the timepiece regulated by the vernal equinox, and rep- resenting its error at the time t by e, and denoting by R, the right ascen- sion of the star, we obtain t+e+c.C+l.L+Z.Z,=R (17) m which, if e, c, /, and Z be regarded as unknown, their values may be found by carefully' observing four stars, whose positions are well known, and which differ but little in right ascension, and considerably in declina- c 1 1 5 . cos o cos(X S) 15 . cos d sin(X-S) 15 .cos 8 ' J APPENDIX If. 265 tion The values of (7, Z, and Z t being computed ill each case from equation (16), we may have f + e '+c.C' +I.L' +z.Z,' =R f t" +e f +c.C" + I.L" +Z.Z/' = R" t'" +e'+c. C'" + I.L'" +z.ZS" =R'" t""+ e f +c . C""+l . L""+z . Z,""=R"" which are sufficient. But as there are always slight errors in the obser- vations themselves, it would be well, where great accuracy is required, to increase the number of these equations, and treat them after the method of least squares. 15. The finding circles. These may indicate zenith distances, altitudes, or polar distances. The rule for adjusting is the same for all. Direct the telescope to the distant horizon, and move it till the. image of some small object appear midway between the double axial wires : clamp the axis, move the index-arm till its level indicates the same read- ing at both ends of the bubble, and note the reading of the vernier. Unclamj and reverse the axis ; bring the image of the same object again between th ; same wires, and clamp the axis ; move the index-arm till the bubble has tho same reading at each end, and again note the reading of the vernier. If the vernier reading be the same as before, the circles are in adjustment ; if not, add the readings together, take the half sum, move the index-arm till the vernier is brought to the reading indicated by this half sum, clamp the index-arm, and bring the air-bubble so as to have the same reading at each end by the adjusting screws of the level. It would be well to verify by repeating the process. It may be, that the finders are gradu- ated from to 360, in which case, if the first reading were a, the second ought to be 360 a. 16. The adjustments in azimuth, collimation, and level being per- fected, the middle normal wire will be a visible representation of that portion of the celestial meridian to which the telescope is pointed ; and when a star is seen to cross this wire in the telescope, it is in the act of culminating. The precise instant of this event being noted by the clock or chronometer, the time of meridian passage is known, and any error in noting this precise time is lessened by the use of the lateral wires of the reticle, as already explained. 17. Besides, these lateral wires increase the chances of securing an observation that might, without them, be lost. It frequently happens that efforts to obtain the time of a body's passing the middle or other wire are defeated by the presence of clouds, or other accidental circumstances, 266 SPHERICAL ASTRONOMY. m which, if the time of passing any one be obtained, that of passing the middle or mean place of the wires, when not equally distant, may be Deduced thus. Let < 2 , 3 , &c., be the times of crossing the several wires in order, hen will (18) n which t m denotes the time of the body's crossing the mean position of ihe wires, and n the number of wires. And (t m t } ).cos8=i } , ' tt z . cos d=i *. cos <= (19) in which S denotes the declination of the body observed, and z,, i 2 , ? 3 . . . ?', the constant intervals of time required for a body in the equator to pass over the distances which separate the several wires from their mean position. Adding equations (19) together, we obtain t m = + -^-r (20) n n cos in which 2 denotes the algebraic sum of the quantities expressed by the letter written after it. By carefully observing a star whose declination is known, we obtain the values of i,, 4, &c. ; and these being tabulated with their proper signs, equation (20) will give the time of a body's passing the mean position from the time of passing one or more of the threads. The Collimating Telescope. 1. In some situations it would not be possible to obtain a distant mark by which to collimate, and a near one could not be used in conse- quence of its image falling too far behind the reticle. In such cases recourse must be had to what is called the collimating telescope. Fif. 28. This is a telescope whose eye-piece is removed, and upon its tube is mounted a small swing- frame, supporting a reflector, by means of which APPENDIX II. 267 sufficient light may be thrown through the telescope to illuminate a pair of cross wires, situated at the solar focus of the object-glass. In this position of the wires, we have, from the principles of optics, these facts, viz. : the rays composing the pencil of light proceeding from any point of the cross, will emerge from the collimator parallel to a line drawn through that point and the optical centre of the lens ; and if the telescope of the transit be directed towards the collimator so as '.o receive these rays, an image of the point in question will appear in its solar focus, and on a line drawn through the optical centre of its object-glass, par- allel to these same rays. The Vertical Collimator. 1. This instrument is used for the double purpose of collimating, and for finding the zenith or horizontal point of circles, used in the meas- urement of vertical angles. It consists of a collimating telescope T mounted in a vertical position upon an annular plate R, of cast-iron, float- ing upon the free surface of mercury, contained, in an annular trough S, also of cast-iron. The annular plate is called the float. The telescope is mounted upon the float in a manner similar to the transit, except that the axis is near- er to the object end. One of the Ys may be elevated or depressed by an ad- justing screw A, while the telescope is turned about its axis by another A 1 ', thus affording the means of giving the line joining the cross wires and the optical centre of the lens a vertical position. L is the lamp, and G the reflector, to catch its light and throw it upon the cross wires at the lower end of the tube. 2. The collimatiny process. J'ake the transit for instance. Level the axis carefully ; turn the telescope in a vertical position ; place th collimator below, and bring the image of the intersection of its cross wires, seen upon the bright ground r, accurately on the intersection of the middle wires in the transit, by means of the adjusting screws of the collimator ; next turn the float in azimuth through 1 80. If the emer- gent rays from the collimator be vertical, the image of the intersection of the collimator's wires will remain stationary, but if not, the image will move in the circumference of a circle ; because, the plane of floatation 268 SPHERICAL ASTRONOMY. remaining the same, the emergent rays from the collimator will preserve their inclination to the horizon unchanged, thus causing the line through the optical centre of the transit's lens, and parallel to these rays, to de- scribe a conical surface. The axis of this cone, which is a vertical line, is the position for the line of collimation. Supposing, then, the image to have changed its position during the semi-rotation of the float, renew the contact of the image and wires ; one half by the adjusting screws of the collimator, and the other half by a motion of the transit and the adjusting screws of the diaphragm of its wires. This process being re- peated once or twice, the adjustment is made. 3. The zenith or horizontal points. Direct the telescope of any circle to the collimator, and bring the image of the intersection of the cross wires in the collimator to the line of collimation ; read the circle, and revolve the float through an azimuth of 180 ; renew the contact of the image line of collimation by moving the circle, if necessary, and read again ; denote the first reading by a, the second by a', and that of the zenith point by z, and we have a-\-a' ,= 180'+ Jt_; and denoting the reading of the horizontal point by k. The Collimating Eye-piece. 4. If now the swing-frame and its reflector be transferred from the collimating telescope to the eye-piece of the telescope of the instrument sup- posed to be vertical over a basin of mercury, this latter telescope becomes its own vertical collimator by reflection, on applying the lamp to the swing reflector. By perforating the swing reflector, and applying the eye behind it, two sets of wires will be seen in the solar focus of the telescope, and the collimating process consists in making the wires of Fig. 25. one of these sets coincident with those of the other, by the joint motion of the telescope and its reticle. The little swing reflector, with a single microscope as an eye- piece, just behind its perforation, to magnify the wires and their images, constitutes the collimating eye-piece. This beautiful little instrument, which has done so much to facilitate the process of collimating an 1 the measure- ment of zenith or nadir distances, is due to Professor J3ohnenberger of Tubingen. APPENDIX II. 269 The Mural Circle. 1 . By means of the transit and a time-keeper, distances are meas- ured on the equinoctial in time ; and by an easy reduction this time is converted into arc. The object of the Mural Circle is to measure dis- tances on the meridian. This instrument consists of a metallic circle A A, varying in diameter from four to eight feet, strongly framed together or cast in one entire piece, and a telescope, of considerable optical power, having a focal length about equal to the diameter of the circle. The circle is firmly attached Fig. 26. to the larger end of a hollow conical-shaped axis at right angles to its plane, which axis is mounted on Ys, placed in an opening through a heavy wall, whose front face is in the plane of the meridian. The gradu- ation is usually, though not always, upon the outer rim, and the readings are made by a pointer and six or more reading microscopes F, mounted upon the face of the wall, at equal distances from each other, around the circle. The telescope is mounted upon the front face of the circle, so as 270 SPHERICAL ASTRONOMY. to move paiallel to the plane of the latter by means of a second axis, which turns freely and concentrically within that of the circle. The axis of the telescope is also conical, and is kept in place and proper contact with that of the circle, by means of a strong nut, which receives a screw cut upon its smaller end, the head of the nut bringing up against the end of the circle's axis. By turning this screw in the direction of its thread, the two axes are brought as closely in contact as may be found desirable. Permanently connected with each end of the telescope is a clamping arrangement, for the purpose of seizing the rim of the circle, and when these are in bearing, the telescope can only move with the circle, and when loose, it may move independently, thus affording the means of meas uring the same angular distance on different parts of the circle. Five clamping and tangent screw arrangements are permanently at- tached to the face of the wall, for the purpose of restricting the motion of the circle to the minute adjustments necessary to complete the contact of the objects observed with the reticle of the telescope, and to secure the instrument till the readings are made and recorded. They are made thus numerous, that one may always be at hand, in the various positions of the observer about the circle ; one of them is shown at E. The proportions of the whole instrument are so adjusted as to throw its centre of gravity on the axis just behind the circle, and between it and the wall, where the axis is received by a stirrup with friction-rollers C (7, the stirrup being connected by rods D D with levers and counter- poising weights, which take the bearing from the Ys. The front Y, or that nearest the circle, is movable in azimuth about a vertical pintle, and that at the smaller end admits of both a vertical and horizontal motion, by means of two sets of antagonist screws. The tube of the telescope is perforated on the side opposite that of the axis to admit the light from a lamp at a short distance in front of the circle ; this light is received upon a perforated reflector within, after the manner of the transit, and thrown to the eye to illuminate the field of view in nocturnal observations. The intensity of the illumination is reg- ulated by square perforations in two sliding plates, placed over the aper- ture in the tube, and so connected with rack and pinion work as to move in opposite directions, on turning a large milled-headed screw near the eye-glass ; one of the diagonals of each square being placed in the direc- tion of the motion of the plates, the figure of the opening will be un- changed, while its size may be varied at pleasure. At P and P are two small t^es, permanently fixed to that of the telescope, and at right angles to its length. They are cut away on one APPENDIX II. 271 side at the middle, and each is closed at one end by a small disk of mother-of-pearl, movable about an axis perpendicular to its plane, and concentric with the tube. Between the disk and middle of the tube is a convex lens, which admits of a motion in the direction of the tube, and by which an image of a small eccentric perforation in the disk is formed about the middle of the cut, and of course on one side of the axis. Aj motion of the pearl causes this image to describe the circumference of a circle, of which the centre is on the axis of the tube. In the opposite end of the tube is a small microscope to view this image. The image is technically called the ghost, being a visible but unsubstantial representa- tion of the perforation. A small metallic style projects from the face of the wall at S, from the end of which may be suspended a plumb-line of fine silver wire, with its bob immersed in a vessel of water or other liquid at the bottom of the wall. The style is so arranged by an adjusting screw as to bring the plumb-line to intersect the axes of the small tubes in the cuts, or to throw it clear of the instrument, at pleasure. In the tail end of the telescope, and at the solar focus of the object- glass, is a reticle, of which the axial wires are parallel to the axis of the circle. An additional wire is driven by a micrometer screw in the direc- tion, perpendicular to the axial wires, while it is also kept constantly par- allel to them. The telescope has a collimating eye-piece, which is used for the same purpose and in the same manner as in the transit. Adjustments. 2. The adjustments are, first, to make the line of collimation per- pendicular to the axis, and, second, to make the axis perpendicular to the meridian. The plane of the circle and tube of the telescope are placed at right angles to the axis by the manufacturer ; the face of the wall is built as nearly in the meridian as possible by the aid of meridian marks ; and the Ys are so placed as to bring the axis, when mounted, nearly per- pendicular to the face, so that the adjustments are approximately made when the instrument is put up. To complete them, begin with 3. The line of collimation. Turn the circle till the telescope is vertical, suspend the plumb-line and bring it by its adjusting screw to co- inci ie with the upper ghost as seen through the microscope : examine the position of the lower ghost ; if it be not on the line, turn the pearl about its axis till it is : clear the line from the instrument, and invert the telescope by revolving the circle through 180; bring the line to the 272 SPHERICAL ASTRONOMY. upper ghost as before, and again examine the lower ghost ; if it be on the line, the axis of the circle is horizontal, but if not, bring it to the line, one-half by the vertical adjusting screws of the circle's axis and half by a revolution of the pearl. When by repeating this process once or twice the axis is made horizontal, put on the collimating eye-piece, and directing the telescope to the trough of mercury at the foot of the pier, and immediately below, move the diaphragm of the cross wires till the wire, which is perpendicular to the axis, coincides with its image the line of collimation will be in a vertical plane, and of course perpendicular to the axis, which is horizontal. Should the telescope have no collimating eye-piece, recourse may be had to the vertical collimator, which is to be used exactly as in the transit. Since reflection takes place in a plane normal to the reflecting surface, the axis may be made horizontal by observing the same star directly, and by reflection from the free surface of mercury. If the time of the star's appearing on the line of collimation in both views be the same, the two positions of the line of collimation will lie m the same vertical plane, and being equally inclined to the horizon, the axis with which they make a constant angle must be horizontal. 4. Axis perpendicular to the meridian. This adjustment may be made by the method pointed out for the same adjustment in the transit ; and when not perfected, the amount of error may be found by the process explained for that instrument. Polar and horizontal points. On the circumference of the circle is a scale of equal parts, each part having an angular value of five minutes. Every twelfth division is numbered, the numbers varying from 1 to 360 inclusive ; these indicate the degrees of the scale; and to facil- itate the reading, the intermediate divisions are also numbered, but in smaller characters. If the reading be known when the line of collimation is either hori- zontal or directed to the pole of the heavens, and the reading be taken when directed upon the centre of any body as it passes the meridian, the difference of the readings will in the first case be the observed meridian altitude of the body, and in the second its observed polar distance. 5. The horizontal point. This is found by means of the collima- ting eye-piece, or vertical coll^mator, by the process indicated at page 268, or as follrws, viz. : having carefully ascertained the value of a revolution of the micrometer in the eye-piece of the telescope, and the reading of its divided head when the movable wire is coincident with that parallel to he axis, set the telescope nearly in the position at which a star would APPENDIX II 273 appear by reflection on the stationary wire ; clamp the circle and record the reading of the index and microscopes ; when the star is at a conve- nient distance from the meridian wire, bisect it by the movable wire with- out moving the circle, and note the time accurately. Unclamp the circle, and bring the star by direct view accurately on the stationary wire, by turning the whole circle about its axis ; again note the time, and record the reading by the index and microscopes. Denote by R the first read- ing, by D the second, and by m the angular value of the distance between the fixed and movable wire, as indicated by the micrometer ; then, if the star had been observed accurately on the meridian, would the reading of the horizontal point be R m-\- D Fig. 27. since the star must appear as far below the horizon by reflection as it actually is above it. But as the star cannot be taken at the same instant in both positions of the instrument, the readings R and Z>, taken as above indicated, must be reduced to what they would have been if taken on the meridian. 6. This correction will now be explained. Ler. S' S S" be the small diurnal circle of the star ; P M S' an arc of the meridian ; tltf position of the star when observed on the intersection of the axial and one of the side normal wires ; MS C the arc of a great circle, of which the axial wire is a portion. The point M will be that to which the line of collimation is actually directed, and S' is that in which the star will reach the meridian ; the arc M S f is, therefore, the reduction to the meridian. Make P MP S = hour angle of star ; d = P S polar distance of star ; y = P M = polar distance of line of collimation ; x = M S' = reduction to meridian. Then in the triangle MP S, right-angled at M, sin y cos d cos P = tan y . cot d = cos y ' sin d ' and subtracting this from 1 = 1, 18 274 SPHERICAL ASTRONOMY. we have, after reducing, and replacing 1 cos P by 2 sin 8 \P, cos y . sin a Tlie observation being made very near the meridian, P and dy will b very small, and hence 2sm 2 JP = 2.(JP.sin I") 2 = P 2 . sin 8 1"; sin (d y) = sin # = rr . sin 1" ; sin c? . cos y = ^ sin 2 c?, very nearly. which in the above equation give, after reduction, a? = Jsin 2c?.P 2 .sin 1", in which P is expressed in seconds of arc. To express it in time, make P=15 P b and we shall finally have OOK x=~.nn2d.Pf.wnI", P, denoting the number of seconds of time in the hour angle of the star. If, now, the numbers on the circle be supposed to increase in the direc- tion from the pole to the zenith, and the observed reading be denoted by R, then, since the line of collimation is nearer the pole than tht- place of culmination of the star, will the true reading be 225 R x = R -- -sin 2d. P, 2 sin 1" . . . . (21) for all stars whose declinations are of the same name as the latitude of the place, and above the pole, and OOK R + x=:R+--sm2d.P*sml" . .... (221 for all stars below the pole, or whose declinations are not of the same name as the latitude. 7. The interval P is obtained from the indications of a time- keeper. This usually runs too fast or too slow. To get the true from the indicated interval, suppose the time-keeper to gain or lose a seconds du- ring one revolution of the earth upon its axis. Ifenote by A. the number of sidereal seconds in the time of this revolution, and by t the true interval Fouht then will APPENDIX II. 275 A a : A : : P : t, t- A P- P- "A~a fl .a' A in which P is the indicated interval. Developing the coefficient of P, and limiting the series to the first power of , because a is usually a small number of seconds, we have or replacing A by its value 86.400, t (1 T . 000012 a) P = a . P, in which a = 1 qp . 000012 a. Substituting aP for P, in equations (21) and (22), and making i = a 2 = 1 ^: . 000023 a, there will result for the true reading OOPi J K= F --.t.P*.sm2c* sin 1" ..... (23) 8. Denote by D and J2 the readings of the circle by the direct and reflected views ; by x and x' the corresponding reductions to the me- ridian ; by m the small difference observed between the angle of incidence and reflection, and by H the reading of the horizonta. point ; then will and _ _ m xx ~~~ ~~~ 2 " ~T~ 9. Value, in arc, of units on the screw-head connected with the movable wire. Run the movable wire to one edge of the field of view, say the upper, and bring it by a motion of the circle upon some well- defined and distant object ; read the circle and micrometer ; run the wire to the opposite or lower edge of the field, and by a motion of the circle bring the wire to same object again ; read the circle and micrometer as before, and divide the difference of the circle readings, reduced to seconds; by the difference of the micrometer readings, expressed in units of the screw-head ; the quotient will be the value sought. Or, Invert the telescope over a basin of mercury, by moving the circle, and 27-6 SPHERICAL ASTRONOMY bring the image of the movable wire, supposed at one edge of the field, to coincide with the wire itself; read the circle and micrometer: move the wire to the opposite edge, and turn the circle till the wire and its image again coincide, and read as before ; divide the difference of the circle readings, reduced to seconds, by the difference of the micrometer readings expressed in units of its screw-head ; the quotient will be the value sought. Altitude and Azimuth Instrument. \ . This instrument, as its name indicates, is employed in the measurement of vertical and horizontal angles. It has two graduated circles and a telescope. The planes of the circles are at right angles to each other ; one called the azimuth circle, being connected with a tripod, by which it is levelled and kept in a horizontal position ; while the other, called the altitude circle, is mounted upon a horizontal axis, with which the telescope is also united, after the manner of the transit. To the centre of the tripod A A is fixed a vertical axis, of a length equal to about the radius of the circle ; it is con- cealed from view by an exterior cone B. On the lower part of the axis, and in close contact with the tripod, is centred the azimuth circle (7, which admits of a horizontal circular motion of about three degrees, for the ' purpose of bringing its zero ex- actly in the meridian; this is effected by a slow moving- icrew, the milled head of which is shown at D. This motion should, however, be omitted in instruments destined for exact work, as the bringing the zero into the meridian is not requi- site, either in astronomy or sur- veying : it is, in fact, purchasing A a convenience too dearly, by introducing a source of error APPENDIX II. 277 not always trivial. Above the azimuth circle, and concentric with it, is placed a strong circular plate E, which carries the whole of the upper works, and also a pointer, to show the degree and nearest five minutes to be read off on the azimuth circle ; the remaining minutes and seconds being obtained by means of the two reading microscopes F. This plate, by means of the cone B, rests on the axis, and moves concentrically with it. The conical pillars H support the horizontal or transit axis /, which, being longer than the distance between the centres of the pillars, the pro- jecting pieces c, fixed to their top, carry out the Ys a, to the proper dis- tance, for the reception of the pivots of the axis ; the Ys are capable of being raised or lowered in their sockets by means of the milled-headed screws 6, for a purpose hereafter to be explained* The axis, with its load; is prevented from pressing too heavily on its bearings, by two friction* rollers, on which it rests ; one of these rollers is shown at e. A spiral spring, fixed in the body of each pillar, presses the rollers upward, with a force nearly a counterpoise to the superincumbent weight ; the rollers on receiving the axis yield to the pressure, and allow the pivots to find their proper bearings in the Ys, relieving them, however, from a great portion of the weight. The telescope K is connected with the horizontal axis, as before re- marked, in a manner similar to that of the transit instrument. Upon the axis, as a centre, and in contact with the telescope on either side, is fixed the double circle J. The circles are united by small brass bars ; by this cir- cle the vertical angles are measured, and the graduations are cut on a narrow ring of silver, inlaid on one of the sides, which is usually termed the face of the instrument : a distinction essential in making observations. The clamp for fixing, and the tangent-screw for giving a slow motion to the vertical circle, are placed beneath it, between the pillars H, and attached to them, as shown at L. A similar contrivance for the azimuth circle is represented at M. The reading microscopes for the vertical circle are supported by two arms bent upward near their extremities, and attached to one of the pillars. The projecting arms are shown at N and the microscopes above at 0, the latter admitting of a slight motion by means of antagonistic adjusting screws independently of the sup- porting arms. A reticle consisting of five equidistant axial and as many equidistant normal wires, is in the principal focus of the object-glass. The illumina- tion of the wires at night is by a lamp, supported near the top of one of the pillars at rf, opposite the end of one of the pivots of the axis, which, leino perforated, admits the light to the centre of the telescope tube, 278 SPHERICAL ASTRONOMY. where, falling on a diagonal reflector, it is reflected to the eye, and illu mines the field of view. The vertical circle is usually divided into four quaarants, each num- bered 1, 2, 3, &c.; up to 90, and following one another in the same order of succession ; consequently, in one position of the instrument alti- tudes are read off, and with the face of the instrument reversed, zenith distances ; and an observation is not to be considered complete till the object has been observed in both positions. The sum of the two readings will always be 90, if there be no error in the adjustments, in the circle itself, or in the observations. It is necessary that the microscopes and the centre of the circle should occupy the line of its horizontal diameter ; to effect which, an up- and-down motion, by means of the screws 6, is given to the Ys. A 'spirit-level P is suspended from the aims which carry the microscopes : this shows when the vertical axis is set perpendicular to the horizon. A scale, usually showing seconds, is placed along the glass tube of the level, which exhibits the amount, if any, of the inclination of the vertical axis. This should be noticed repeatedly whilst making a series of observations, to ascertain if any change has taken place in the position of the instru- ment after its adjustments have been completed. One of the points of suspension of the level is movable, up or down, by means of the screw f, fo the purpose of adjusting the bubble. A striding-level, similar to the one employed for the transit instrument, and used for a like purpose, restr upon the pivots of the axis. It must be carefully passed between the radial bars of the vertical circle to set it up in its place, and must be re- moved as soon as the operation of levelling the horizontal axis is per- formed. The whole instrument stands upon three foot-screws, placed at the extremities of the three branches which form the tripod, and brass cups are placed under the spherical ends of the foot-screws. A stone pedestal, set perfectly steady, is the best support for this as well as the portable transit instrument. Adjustments. 2. These have for their object to make, 1st, the azimathal axis per- pendicular to the horizon ; 2d, to make the axis of the vertical circle horizontal ; 3d, to place the vertical circle at such a height that its mi- croscopes shall point to the opposite extremities of a horizontal diameter ; 4th, to make the line of collimatiori perpendicular to the axis of the alti- tude circle, and horizontal when the reading of the vertical circle is zero. 3. The vertical axis. Turn the instrument about, until the spirit- APPENDIX 11. 279 level P is lengthwise in the direction of two of the foot-screws, when by their motion the spirit-bubble must be brought to occupy the middle of the glass tube, which will be shown by the divisions on the scale attached to the level. Having done this, turn the instrument half round in azi- muth, and if the axis is truly vertical, the bubble will again settle in the middle of the tube ; but if not, the amount of deviation will show double the quantity by which the axis deviates from the vertical in the direction of the level ; this error must be corrected, one-half by means of the two foot-screws, and the other half by raising or lowering the spirit-level itself, which is done by the screw represented at/. The above process of rever- sion and levelling should be repeated, to ascertain if the adjustment has been correctly performed. Next turn the instrument round in azimuth a quarter of a circle, so that the level P shall be at right angles to its former position ; it will then be over the third foot-screw, which may be turned until the air-bubble is again central, if not already so, and this adjustment will be completed ; if delicately performed, the air-bubble will steadily remain in the middle of the level during an entire revolution of the instrument in azimuth. These adjustments should be first performed approximately, for if the third foot- screw is much out of the level, it will be impossible to get the other two right. The vertical axis is now adjusted. 4. The axis of the vertical circle. This adjustment is performed exactly as in the transit, by means of the striding-level. 5. Height of the vertical circle. The last adjustment being made, bring the microscopes to their zeros, and turn the vertical circle slightly, the striding-level being still mounted, till some one of its divisions be brought to the cross wires of one of the microscopes. Examine the other microscope, and if its cross be not on or near the division of the circle, 1 80 distant from the first, depress or elevate the circle by the milled screws b till it is, keeping the axis horizontal by means of the level ; this will give a sufficient approximation to bring the error of adjustment within the range of the adjusting screws which move the microscopes indepen- dently of their supporting arm. Recourse must now be had to these screws, by turning which in the direction indicated by the relative posi- tion of tha circle division in question and the cross wires, the adjustment is perfected. 6. The line of collimation. As the vertical circle is not, like the mural, generally used as a differen- tial instrument, but in the measurement of absolute altitudes or zenith distances, it is not only necessary that the line of collimation shall be per* 280 SPHERICAL ASTRONOMY. pendieular to the transit axis, but also that it shall be parallel to the radius of the graduated circle drawn to the zero of its scale. Let x denote the angle made by the line of collimation with the plane normal to the transit axis, which angle is usually very small, and a the reading of the azimuth circle, when the -telescope is pointed to some well- defined object in or near the horizon. If the line of collimation lie on the side of the normal plane, towards the zero of the circle, the true reading will be sensibly equal to a x, if there be no other error of adjustment. Now revolve the instrument in azimuth 180, bring the telescope again on the object, and denote by a! the new reading ; the tine reading now will be '+ v, the difference of these true readings is obviously a semi-circumference, whence a _ a' <2 X = 180; and a a'- 180 *=- " -' and the true reading in the second position becomes a - a '\ 80 Again, denote by y the small angle which the line of collimation makes with the plane passing through the axis of the vertical circle and that zero of this latter circle nearest the line of collimation, and suppose the line of collimation to lie above this plane when the telescope is directed to the same object, as before. Let b denote the apparent altitude, supposing the circle in the position to mark altitudes ; the true altitude is sensibly equal to 6-hy; turn the instrument in azimuth 180, and bring the telescope again on the object ; the line of collimation will now be below the plane of the axis and zero, but the circle now indicates a zenith distance &', whence the true zenith distance is adding these measures together, we have APPENDIX II 281 &-f6'+2y = 90 90 -(6 +6') y = - and the true zenith distance becomes , W-(b + b>) ~2~ Whence to adjust the line of collimation we have this rule, viz. : Direct the telescope to some well-defined and distant object, not far from the horizon, and bring its image to the intersection of the middle wires ; record the reading of the azimuth and vertical circles ; turn the instru- ment in azimuth 180, bring the line of collimation again on the object, and record the new readings of the circles ; subtract from the difference of the azirnuthal readings 180, divide by 2, and add (algebraically) the quotient to the last azimuthal reading for a new reading in azimuth. Add the two readings of the vertical circle together, subtract the sum from 90, and add half the difference to the last reading for a new reading on the vertical circle. Set the circles to these new readings, clamp, and by the adjusting screws of the reticle bring the line of collimation to the object, and the adjustment is made ; it should be verified, however, by repetition. 7. To make the normal wires perpendicular to the transit axis, proceed as in the case 01 the transit instrument, viz. : Move the diaphragm about in its own plane, till the image of some object appears to run accti rately along some one of the wires, say the middle one, while the tele- scope is turned about its axis. 8. The altitude and azimuth instrument is regarded by many as the most universally useful of all astronomical instruments. It is portable and accurate. When used in the meridian, it may perform the work of the transit and mural circle, though with somewhat diminished accuracy. But its principal merit consists in the ease with which it may be moved in azimuth without impairing its measurement of altitudes and zenith dis- tances. 9. The instrumental bearing of an object is the angle indicated by the reading of the azimuth circle when th'e centre of the object is ap- parently on the line of collimation. From the instrumental, the true bearing, or true meridian, is found by a process to be explained hereafter. 10. To find the altitude and instrumental bearing of an object at any instant, it is only necessary to make the object pass the line of colli- mation by turning both tangent-screws as it moves through the field of vifw, and to note the time of passage, ana v< ad the circles. 282 SPHERICAL ASTRONOMY. 11. The altitude and time, or the instrumental bearing and time, are the elements more commonly observed in the case of celestial objects. 12. To obtain the altitude and time. With the circles undamped, direct the telescope, which it will be remembered inverts, so as to bring the image of the object in the lower or upper part of the field of view, as the body may be rising or setting ; clamp the circles, and by the tangent screw of the azimuth motion, bring the image to the middle normal wire, and keep it there till it passes all the axial wires, carefully noting the time of its passing each, and also noting the indications of the level be- fore it passes the first and after it passes the last one. Now read the vertical limb, unclamp, and, by an azimuthal motion, reverse the face of the vertical circle without unnecessary loss of time, and go through the same operation as before. Reduce the vertical readings to the same de- nomination of altitude or zenith distance, correct them by applying the level readings, and take half the sum for the altitude or zenith distance, as the case may be. Add the times together, and divide the sum by the number of recorded times for the corresponding time. 13. To find the instrumental bearing and time, direct the telescope as before, and clamp ; with the tangent-screw of the vertical motion, bring the image of the object to the middle axial wire, and keep it there till it passes all the normal wires, on each of which record the time. The read- ing of the azimuth circle will give the instrumental bearing, and a mean of all the times will give the corresponding time. All of this supposes that the object's change in altitude and azimuth is uniform ; and although this is not strictly true, it is nevertheless so nearly so for the short time its image is in the field of view, that the error will be inappreciable during the interval required for a single set of ob- servations. The Equatorial. 1. The object of the equatorial or parallactic, as it is frequently called, is to support a telescope, generally of great size and optical power, in su;h manner as to give to the observer the means of directing it with ease io any part of the heavens, and to measure at once the apparent hour angle and polar distance of a heavenly body. In the principles of its con- struction, it is like the altitude and azimuth instrument, but differs from it in the position of its axes, which, instead of being vertical and horizontal, are, when in position, respectively perpendicular, and parallel to the plane of the equinoctial. The first is called the polar, the second the declina- tion axis. It has two graduated circles, one securely attached to each APPENDIX Ii. 283 axis ; the plane of one, viz., that attached to the polar axis, is parallel, ant) the other perpendicular to the equinoctial. The first is called the Aowr, and the second the decimation circle. By a motion of the polar axis, to which the supports of the declination axis are attached, the declination circle may be made parallel to any assumed declination circle of the celes- tial sphere. The polar axis, always much loaded, is, in low latitudes, con- siderably inclined to the horizon, and the practical difficulty of supporting it has given rise to a variety in the form of the instrument. That repre- sented in the figure is the one now most generally used, and it is intro- duced here on that account. The principle is the same in all. The supporting-stand is shown at H, H, H. It is made either of a strong frame of wood-work, or is cut from a solid block of stone. B is a plate of metal, firmly secured to the stand, the surface of contact being parallel to the axis of the heavens. Upon this plate the instrument is mounted. The polar axis is seen at /. It is of steel, and revolves in two cylindrical collars near the extremities, and the lower end, being rounded off and highly polished, rests upon a steel plate attached to a bearing- piece K. To the lower end of this axis is attached the hour-circle 72, which is either graduated into hours, minutes, and seconds, or into degrees and the usual subdivisions, at the option of the person ordering the instrument. The verniers, or reading microscopes, and tangent-screw arrangement, are supported by pieces connected with the plate B. Tho declination axis revolves in a metallic tube M, which forms a part of the frame-work se- cured to the top end of the polar axis. To one end of the declination axis is attached the declination circle P, which is graduated so as to read polar distances or declinations suppose the former, it micrometers and tangent- screw being mounted upon pieces projecting from the extremity of the tube M, and to the other end, which projects slightly beyond the frame- work, is attached the telescope at a point nearer the eye-end than the middle. The excess of weight towards the object-end is, in the mounting by Mr. Henry Fitz, of New York, compensated by a counterpoise cylin- drical lever within the tube of the telescope, and so arranged in bearing as to counteract all tendency in the tube to bend. Attached to the end of the declination axis, is a counterpoise weight 0, the office of which is to throw the centre of gravity of the entire movable part of the instrument in the polar axis near its upper end, where it is received by a pair of fric- tion-rollers. At C is a box containing a system of wheel-work, so connected with the polar axis as, by the lid of weights and a centrifugal governor, to give 281 SPHERICAL ASTRONOMY Fig. 29. it a uniform motion of rotation. The velocity of rotation is regulated by a vertical motion of the axis of the governor, whose balls in their retro- cession and increasing velocity, force a pair of rubbing surfaces against the interior of an inverted conical box : the moment of the friction thence arising equilibrates that of a descending weight, and the motion become* APPENDIX II. uniform. By elevating the axis of the governor, the motion is acceler- ated ; by depressing the axis, it is retarded, and thus the velocity of rota- tion may be made equal to that of the earth about its axis, in which case a star in the field of view will be kept there ly the instrument itself, the effect being the same, abating refraction, as though the earth were at rest, 2. With a divided object-glass for the telescope, to be explained presently, or with the position micrometer, the equatorial is mostly used as a differential instrument, and particularly when the observer is pro- vided with a very full and accurate catalogue and map of the stars, which serve as points of reference. Whenever it is possible to bring a known object into the field of view with one that is not known, the place of the latter is found by measuring its bearing and distance from the known object. 3. To measure directly the hour angle and polar distance of an object with the equatorial, requires the parts of the instrument to be in perfect adjustment among one another, and its polar axis to be parallel to the axis of the earth. For these adjustments and a full analysis of the equatorial. Analysis of the Equatorial. The true instrumental position of an object is that indicated by an in- strument in perfect adjustment within itself. The apparent instrumental position is that actually indicated by an instrument whether in adjustment or not. When the several parts of an instrument are adjusted with respect to each other, these two positions are the same. The instrumental hour angle of an object, is its angular distance from a vertical plane passing through the polar axis, estimated upon the hour circle. Its instrumental declination is its angular distance from a plane perpen- dicular to the polar axis, estimated upon the declination circle; and its in- strumental polar distance, its angular distance from the polar axis. The line of collimation should be perpendicular to the declination axis, and the latter perpendicular to the polar axis. The index of the hour cir- cle should stand at the zero of the scale when the line of collimation is parallel to the vertical plane of the polar axis, and, supposing the instru- ment to read polar distances, the index of the declination circle should be at the zero of its scale, when the line of collimation is parallel to the polar IDA, 286 SPHERICAL ASTRONOMY. Supposing none of the conditions to be fulfilled, the apparent instru- mental position of an object will differ from the true, and the first thing to be done is to find the latter from the former, when the error in each of the above particulars is known. To do this, we will premise that the equatorial may be regarded as an universal transit instrument, whose horizon is the equinoctial, and zenith the pole. The formulae of reduc- tion applicable to the transit will apply at once to the equatorial by making therein the symbol for the latitude 90; in which case we shall have for the difference between the true and apparent instrumental hour angle in arc, the sum of the last three terms of Eq. (15), viz., c cos (X #) sin (X } - | % - ^ ~\ & ^ COS COS COS which reduces, by making X = 90, and replacing S by 90 - - tf, to c . cosec * + I cot = * -f-A-^+X. cos (tf 9), in which n = c . cosec * -|- / . cot *, n'= c . cosec ^ + ^ . cot K r Subtracting the first from the second, the third from the fourth, transpo- sing, and making, after eliminating s r and * by their equals t' a, t a, 2 = (t r - *) - (*' _ , . . . (t) whence o-' + er- 12 h n ,,. - -- --- tan-' . and from equations (A) we have -12h. su To find Ati. Observe a star before its culmination in the hour angle 360 tf, and at an interval after its culmination in the hour angle tf', such, that 360 tf and tf' shall be equal, or very nearly so, without re- versing the declination circle ; Eq. (d) will then give 24 h - s = 24 h tf + A tf + X cot if sin (360 tf + 9) n, s' = 0" + A tf + X cot * . sin (*' (p) + n. Adding and reducing, s' s = tf' tf + 2Ao' + Xcot'7r'. [sin (^ 9) sin (tf + 9) ] ; writing sin (tf (p) for sin (tf 7 (p), to which it is sensibly equal, we have, after developing the last term, reducing, and replacing * and s f by their equals t a and t' a, A tf = - ^ t 4- X . cot if . (sin 9 . cos tf) . . (m) For a star in or near the equator, we may take cot if = ; or for a star whose hour angle is 90, in which case cos tf = 0, the above value for index error becomes To find A if. Observe the same star twice in quick succession, and in reversed positions of the declination circle ; the readings of the declinatioh circle, in Eq. (e), give p = if + & and because M D and M' D' are parallel, the first member is the inclina- tion of the first incident to the second reflected ray. APPENDIX II. 295 If then the reflector CB were transparent at the point N, the waves of ight from an object at R', would be transmitted through it and coincide in direction with those from R reflected at M and N; and to an eye situated at 0, the objects R and R' would apparently coincide. Two reflectors so mounted as to give the means of reading their inclination to each other, when this coincidence takes place, would give the angular distance ROR' of the objects by simple inspection ; and, with appliances to facilitate the operations of the "observer, constitute a reflecting instrument, which, ac- cording as its arc of measurement is extended to an entire circumference or limited to an arc of 90, 60, or 45, is called a reflecting circle, quadrant, sextant, or octant. The sextant is the more common of the instruments with limited arcs now in use. 2. The annexed figure represents a sextant. It consists of the two plane-glass reflectors C and E seen edgewise; a graduated arc A A, of which the plane is perpendicular to those of the reflectors ; an index-arm F, vernier V, clamp and tangent screw ; a telescope ED, of which the line of collirnation is parallel to the plane of the arc of measurement; col- ored glasses L and K to qualify the light received into the telescope, and a triangular system of frame-work uniting strength with lightness, to sup- port all the parts and render them available. The handle of the instru- ment is represented at H. The arc of measurement is divided into half-degree spaces, which are numbered as whole degrees, and these divisions are subdivided to any de- Fig. 81. SPHERICAL ASTRONOMY. feirable extent consistent with facility of reading. The reflector B, called the index-glass, is covered with an amalgam of tin on the face towards the eye-end of the telescope, and turns with the index-arm about an axis in its own plane, and through the centre of the arc < f measurement, being per- pendicular to the plane of the latter. The reflector (7, called the horizon- glass, is, abating the limited range of the adjusting screws, securely fixed with its plane also at right angles to that of the arc of measurement. Only one-half of this glass is covered, and that half lies nearest the frame of the instrument, the covered face being turned from the telescope. The line separating the covered from the uncovered part of this glass is parallel to the plane of the graduated arc, and at a distance therefrom about equal to that of the line of collimation, being sometimes a little greater and some- times a little less in consequence of a change in the position of the tele- scope, to make the supply of light it receives through the uncovered, equal to that which enters it after reflection from the coated part of the horizon- glass. The position of the telescope is altered by means of a screw and milled nut connected with its supporting ring U. By turning the nut the telescope is thrust from or drawn towards the face of the sextant. This device is called the up-and-down piece. There are usually six or seven colored glasses of different shades, which are so mounted that they can be turned about an axis c or b parallel to the face of the sextant, and be inter- posed or not at pleasure. To facilitate the reading, a small microscope G is attached to a swing movable about an axis a, connected with the index-arm. Two telescopes and a plane tube, all adapted to the ring C7, are packed with the sextant One of these telescopes has a greater magnifying power than the other, and inverts the visible images of objects. The telescopes are provided with colored glasses, which are so mounted as to be easily attached to the eye- end to qualify the light of the sun when that body is observed. Adjustments. 3. The sextant requires three adjustments, , viz. : 1st. To make the index and horizon glasses perpendicular to the plane of the arc of measure- ment. 2d. These glasses parallel to each other when the index is at the zero of the scale. 3d. The optical axis of the telescope parallel to the plane of the arc of measurement. 4. To accomplish the first, move the index to the middle of the arc, then holding the instrument horizontally with the index-glass towards the eye, look obliquely clown this glass so as to see the circular arc by direct view and by reflection at tb') same time. If the arc appear broken, the APPENDIX II. 297 position of the glass must be altered till it appear continuous, by means of small screws that attach the frame of the glass to the instrument. The horizon-glass is known to be perpendicular to the plane of the in- strument when, by a sweep of the index, the reflected image of an object and the image seen directly, pass accurately over each other ; and any er- ror is rectified by means of an adjusting screw, provided for the purpose, at the lower part of the frame of the glass. 5. The second adjustment is effected by placing the index or zero point of the vernier to the zero of the limb ; then directing the instrument to some distant object (the smaller the better), if it appear double, the ho- rizon-glass must, after easing the screws that attach it to the instrument, if there be no adjusting screw for the purpose, be turned around a line in its own plane and perpendicular to that of the instrument, till the object ap- pear single; the screws being tightened, the perpendicular position of the glass must again be examined. The adjustment may, however, be rendered UD necessary by correcting an observation by the index error. The effect of this error on an angle measured by the instrument is exactly equal to the error itself: therefore, in. modern instruments, there are seldom any means applied for its correction, it being considered preferable to determine its amount previous to observing, or immediately after, and apply it with its proper sign to each observation. The amount of the index error may be found in the following manner : clamp the index at about 30 minutes to the left of zero, and looking towards the sun, the two images will ap- pear either nearly in contact or overlapping each other ; then perfect the contact, by moving the tangent-screw, and call the -minutes and seconds denoted by the vernier, the reading on the arc. Next place the index about the same quantity to the right of zero, or on the arc of excess, and make the contact of the two images perfect as before, and call the minutes and seconds on the arc of excess the reading off the arc; half the differ- ence of these numbers is the index error; additive when the- reading on the arc of excess is greater than that on the limb, and subtractive when tbft contrary is the case. Example. i a Reading on the arc ... 31 56 " off the arc ... 31 22 Difference . . . Index error . . . 298 SPHERICAL ASTRONOMY. In this case the reading on the arc being greater than that on the art of excess, the index error, = 17 seconds, must be subtracted from all ob- servations taken with the instrument, until it be found, by a similar pro- cess, that the index error has altered. One observation on each side of zero is seldom considered enough to give the index error with sufficient exactness for particular purposes : it is usual to take several measures each way; "and half the difference of their means will give a result more to bb depended on than one deduced from a single observation only on each side of zero." A proof of the correctness of observations for index error is obtained by adding the above numbers together, and taking one-fourth of their sum, which should be equal to the sun's semidiameter, as given in the Nautical Almanac. When the sun's altitude is low, not exceeding 20 or 30, his horizontal instead of his perpendicular diameter should be measured (if the observer intends to compare with the Nautical Almanac, otherwise there is no necessity) ; because the refraction at such an altitude affects the lower border (or limb) more than the upper, so as to make his perpendicular diameter appear less than his horizontal one, which is that given in the Nautical Almanac : in this case the sextant must be held horizontally. 6. The third adjustment is made by the aid of two parallel wires placed in the common focus of the telescope for the purpose of directing the observer to the centre of the field of view, in which an observation should always be made; these wires are parallel to the plane of the instru- ment, and divide the field of view into three nearly equal parts. The sun and moon are made tangent to each other, when their angular distance is 90 or more, at one of the wires ; the position of the sextant is then altered so as to bring these bodies to the second wire ; if the contact continue, the line of collimation is parallel to the plane of the instrument ; if not, the position of the telescope must be altered by means of two adjusting screws connected with, the up-and-down piece. Artificial Horizon. 7. To measure directly the altitude of any celestial object with the sextant, it would be necessary that the object and horizon should be dis- tinctly visible ; but this is not always the case in consequence of the irreg- ularity of the ground which conceals the hcrizon from view. The observer APPENDIX II. 299 Fig. 33. is therefore obliged to have recourse to an artificial horizon, which consists usually of the reflecting surface of some liquid, as mercury contained in a small vessel A, which will arrange its upper surface parallel to the natural horizon D AC. A ray of light S A, from a star at , being incident on the mercury at A, will be reflected in the direction A E, making the angle SAC=CAS r (AS' being EA produced), and the star will ap pear to an eye at E as far below the horizon as it actually is above it Now with a sextant whose index and horizon glasses are represented at 1 in.l H, the angle SES' may be measured ; but SES'=SA S' A SE, and because A E is exceedingly small as compared with A S, the angle A S E may be neglected, and S E S' will equal SA S', or double the alti- tude of the object: hence one-half the reading of the instrument will give the apparent altitude. At sea, the observer has the natural or sea horizon as a point of departure, and the altitude may be measured directly. 8. Having now gone through the principle and construction of the sextant, it remains to give some instructions as to the manner of using it. It is evident that the plane of the instrument must Fig 88. be held in the plane of the two objects, the angular distance of which is required. The sextant must be held in the right hand, and as loosely as is consistent with its safety, for in grasping it too firmly the hand is apt to be rendered unsteady. When the altitude of an object, the sun for instance, is to be observed, the observer, having the sea-horizon before him, must turn down one or more of the dark glasses or shades, according to the brilliancy of the object; and directing the telescope to that part of the horizon immediately beneath the sun, and Fig. 84. SPHERICAL ASTRONOMV holding the instrument vertically, he must with the left hand slide the index forward, until the image of the sun, reflected from the index-glass, appears in contact with the horizon, seen through the unsilvered part of the horizon-glass. Then clamp, and gently turn the tangent-screw, to make the contact of the upper or lower lirnb of the sun and the horizon perfect, when it will appear a tangent to his circular disk. When an arti- ficial horizon is employed, the two images of the sun must be brought into contact with each other. To the angle read off apply the index error, and then add or subtract the sun's semidiameter, as given in the Nautical Al- manac, according as the lower or upper limb is observed, to obtain the ap- parent altitude of the sun's centre. The Principle of Repetition. 1, By this principle, the invention of Borda, the error of graduation Hii any instrument may be diminished, and, practically speaking, annihilated. Let P Q be two objects which we may suppose fixed, for purposes of mere explanation, and let L be a telescope movable on 0, the common axis of two circles, A ML and a be, of which the former A ML is fixed in the plane of the objects, and car- ries the graduations, and the latter is free- ly movable on the axis. The telescope is attached permanently to the latter circle, and moves with it. An arm OaA carries the index or vernier, which reads off the graduated limb of the fixed circle. This arm is provided with two clamps, by which it can be temporarily connected with either circle, and detached at pleasure. Suppose, now, the telescope directed to P. Clamp the index- arm OA to the inner circle, and unclamp it from the outer, and read orl Then carry the telescope round to the other object Q. In so doing, th. fnner circle, and the index-arm which is clamped to it, will also be carried round, over an arc A, on the graduated limb of the outer, equal to tin angle P Q. Now clamp the index to the outer circle, and unclamp tin. inner, and read off: the difference of readings will of course measure tht angle P Q] but the result will be liable to two sources of error that of graduation and that of observation, both of which it is our object to get rid of. To this end transfer the telescope back to P, without unclamping the arm from the outer circle; then, having made the bisection of P, APPENDIX II. 301 clamp Jie arm to b, and unclamp it from J5, and again transfer 1.1m tele- scope to Q, by which the arm will now be carried with it to G Y , over a second arc B C, equal to the angle P Q. Now again read oft'; then will the difference between this reading and the original one measure twice the angle P Q, affected with both errors of observation, but only with the same error of graduation as before. Let this process be repeated as often as we please (suppose ten times) ; then will the final arc AE C M read oft' on the circle be ten times the required angle, affected by the joint errors of all the ten observations, but only by the same constant error of graduation, which depends on the initial and final readings off alone. The Reflecting Circle. 1. The use of this instrument is, in general, the same as that of the sextant ; but when it unites, as it often does, to the catoptrical prin- ciple of this latter instrument, the principle of repetition, it becomes, in the hands of a skilful observer, one of the most refined and elegant of the portable implements in the service of astronomy. This form of the instrument is represented in the annexed figure. The arc of measurement, which is extended to the entire circum- ference, is divided into 720. equal parts, and, for the reaton explained in the account of the sextant, these parts are numbered as whole de- grees, the subdivisions being continued to any desirable degree of mi- nuteness. The circle is mounted upon two concentric axes, which may move in- dependently of each other, and also of the circle. Upon one end of the 302 SPHERICAL ASTRONOMY. central axis is mounted a reflector E, similar to the index-glass of the sextant^ and upon the other an arm A C, in the position of a diameter of the circle. Upon the corresponding ends of the other axis are mounted a system of frame-work and a second arm B I). This frame-work sup- ports a second reflector F, similar to the horizon-glass of the sextant, a telescope If, colored glasses L and L\ and the handles /, - log (1 + *) + C; and making # = 0, in which case D = D 4 and g = Z, we have log sin Z =XZ>, -f (7; and by subtraction, sin s or log -; ^ = log t sin tu whence But, Eq. (4), j -v j r L X sin 2 . dD dr = \dD tan 2 = -- Vl sin 8 z and substituting the value of sin s above, If the law which connects the varying density D with the height x be given, one of these variables may be eliminated and the integration per- formed. But in a practical point of view this is not necessaiy ; for X is known to be a very small fraction, as is also the greatest value of rr, the latter not exceeding 0,01931, being the height of the first stratum of air that has sensible action upon light, divided by the radius of the earth, or 77 miles divided by 4000 miles. Developing the factors e~* ' * ' and e 2X (D D)^ ne gi ec t,i n g the second and higher powers of X and x, and also the term of which X sin 8 Z is a factor, which may be done without sensible error when Z does not exceed 80, wi; find 308 SPHERICAL ASTRONOMY. X . sin ZdD X sin Z . dD dr = dr = Vl + 2x sin 2 Z Vcos 2 Z _^L^ = x tan Z . (1 - x sec 2 Z) dD; whence r = X tan zf(dD - sec 8 ZxdD) and performing the integration, that of the last term by parts, r = X tan Z \D sec 2 Z (Dx but if 7^ denote the height of the mercurial column at any stratum of air above the observer, D u the density of the mercury, and g the force of gravity regarded as constant, then will and r = X tan Z [D - sec 2 Z (Dx - D ti h) + (7J; and from the limit x = 0, where D = D' and h = h,, to the limit x = height of the entire atmosphere, where D = 0, r = 0, and A = 0, we find r = X tan Z . D' (\ - h . ^ sec 2 Z\. Taking the density of Mercury as unity, we have the mean value of The mean value of h is found from the proportion, miles inches 4000 : 29.6 : : 1 : h: which will give for the coefficient of sec 2 Z, h.Qi = 0.0012517. Also, if D, be the density of air when the thermometer is 50, and the ba- rometer 30 inches; and we take a = 0.00208, and /3 = 0.0001001, the coefficients of expansion for air and mercury respectively, then, Analytical Mechanics, 245, A 1-f (50-Q./3 ' ' 30 ' 1 -f (t - 50) . a ' in which t denotes the actual temperature of the air and mercury supposed the same, and h the height of the barometer. Hence APPENDIX III. 3Q9 "*- - 0012517 sec ' * 8 Had the second power of x been retained in Eq. (7), then would r = X D . . 1 H 5 ~^ ^ - ^n Z\ 1-0.0012517 sec" Z+ 0.00000139 30 l-f-(< 50) a \ cos*Z the last term of which, within the limits supposed, is insignificant. Make s !tir_"4f ten z (1 - - 0012517 sec ' * } (9) and we have r = \D,u ......... (10) Denote by z and z' the greatest and least observed zenith distances of a oircumpolar star, r and r' the corresponding refractions, and c the zenith distance of the pole ; then will c = In like manner, if 0, and z/ be the greatest and least zenith distances of another circumpolar star, r, and r/ the corresponding refractions, 2 Equating these values, replacing the refractions by the values given in Eq. (10), we find The indications of the barometer and thermometer being substituted in Eq. (9), give u, u', u t , and /, and therefore the value of X J9,. Numerous and careful observations make XD y = 5 7 ".82, which substituted in equa- tions (8) and (8)', give the refraction for every observed zenith distance, temperature of the air, and height of the barometer. 310 SPHERICAL ASTRONOMY. APPENDIX IT. SHAPE AND DIMENSIONS OF THE EARTH. Let A MPjA' represent a meridional section of the terrestrial ellipsoid, M the place of the spectator, B the "north pole 7 j^ of the earth, C its centre, Z the zenith, #*' HMH' a parallel to the rational horizon and tangent to the meridian section at M^ A' A the intersection of the equator by che meridian plane. Make / = the angle M GA = PKH= latitude of M ; A = CA, the equatorial radius ; B = CBj the polar radius. Then, /eferring the curve to the centre and axis, its equation is A? y* + -B 2 a? = A*B* (a) & \ / the equation of the tangent line HIT, A*yy' + B*xx f = A*E* (b) and the equation of the normal at M, A*y'(x-x')-J?x'(y-y') = (c) in which x f and y' are the co-ordinates of M. Denote the angle MTCby T, then from Eq. (b) we have * tan T=^-,\ A*y but T 90 I, whence B*x' tan / = A*y f (d) Also, denoting the eccentricity by , we have Substituting x' y f for xy in Eq. (a), combining the resulting equation with Eq. (c?), and eliminating B by means of Eq. (), we find APPENDIX IV. A cos / 311 " VI - e 2 sin 8 / , A (1 e z ) . sin y == TZ==^=Z=T- V 1 e 2 sin 2 I Differentiating the first, regarding x and I as variable, we have (1 - e* sin 8 1)% but, designating by s the linear dimension of any portion of the arc of the curve, we have for the projection of the element ds on the axis of x, ds . cos T = ds . sin I; and since a? is a decreasing function of the latitude, dx' = ds . sin /; which substituted in Eq. (g) gives ds = A . - .... . (k) (1 - e> sin 2 Z)2 For any other latitude I', we have (1 - e* sin 8 J')* ' dividing the first by the second, making and solving with respect to e 1 , re find . , ds-ds' * dssm*l-ds' sin 8 /' ' From Eq. (h) we have and from the well-known property of the ellipse, B = A Vl & (&) Making ds = c, ds' = c', / = l m , V = J ; m , we have equations (10) and (11) of the text SPHERICAL ASTRONOMY. Denoting by R the radius of curvature at any point of the meridian, we have dxd*y ' finding the values of dv, dy, and d*y from Eqs. (/), and substituting above, there will result (1 - e* sin 2 I)* Then ZR : 360 : : j3 : 1; whence in which /3 denotes the linear dimension of one degree of latitude. Denoting by p the radius of the earth in any latitude /, we obtain by squaring and adding Eqs. (/), Every section of the terrestrial spheroid through the centre is an ellipse of which the semi-transverse and semi-conjugate axes are respectively A and p, / being the latitude of the extremity of the conjugate axis. Denoting by e f the eccentricity of the elliptical section, we have 2 _ ^ 2 -p 8 _ e 8 (1 - e 2 ) sin* I ' = A 2 1 -e* sin 2 I this value of ef substituted in Eq. (I) after making therein / = 90, and denoting by (3, the length of a degree on the section perpendicular to the meridian in the latitude /, R A V l ~ e * sin2 1 ( \ P/ ~360 V: Y l-e 2 (2-e 3 )sin 2 Z The value of the radius of the parallel of latitude is given by that of a?', Eqs. (/) ; and denoting by a the linear length of a degree of longitude on this parallel, we have 2 tf 2 if cos I a = . a = . A . .... (o) 360 360 v/l-e a sin 8 / APPENDIX V. 313 Dividing both members of Eq. (a) by A* B*, making A= 1 and B =^ that equation becomes y T + **=l ......... (f) Differentiating, we find but the angle at Jfin the evanescent triangle m Mh is equal to the angle at G = I' in the triangle MQD\ and denoting in future the central lati- tude M CD by Z, we have -*>, dy X whence tan I 7* tan /' ....... (q) Making A = 1, B =y, and eliminating e 2 from Eq. (m) by the relation = 1 y 8 , we have P = APPENDIX Y. EARTH'S ORBIT. The sun's attraction for the earth varies inversely as the square of the distance. The earth describes, therefore, an ellipse about the sun, having the latter body in one of its foci. By Eq. (266), Analytical Mechanics, we have da. 2c in which a denotes the angle which the radius vector of the earth makes with any assumed axis, r, the radius vector, c the area described by the lat* ter in a unit of time, and t the time. 314 SPHERICAL ASTRONOMY. Also, Eq. (277), Analytical Mechanics, a(l-e*) = ^ ...... . (6) in which a is the semi-transverse axis of the earth's orbit, e its eccentricity, and k the intensity of the sun's attraction on the unit of mass of the earth at the unit's distance. The polar equation of the ellipse is in which V is the true anomaly, estimated from the perihelion. Eliminating r and c from Eq. (a) by means of Eqs. (6) and (c), we have developing the factors of the second members by the binomial formula, and neglecting all the terms involving the powers of e higher than the sec- ond, we have V* ^ . d t = (1 - | ) (1 - 2* cos F + 3e 2 cos 2 F- &c.) da. s and because cos 8 F = + cos2 F, and, 201, Analytical Mechanics, in which T is the periodic time, and m the mean daily motion of a point on the radius vector at the unit's distance from the sun ; whence we have = da 2ecos VdV+l e* cos 2 Vd2 V &c.; and by integration, mt+ C=a 2esiu F+f e 2 sin 2F &c. Making F = 0, and estimating a from the line through the vernal equinox, we have mt p + (7= a,; APPENDIX V 315 in which a, f 's the longitude of the perihelion, and t p the time from peri- helion passage. Whence, by subtraction, m (t t p ) = a a p 2 e sin V + f e* sin 2 V &c. . (e) but a-a p =F; whence w (* t p )= V 2esin F+f e 2 sin 2 7 &c. . . (g) in which w (J ^) is the mean anomaly, being the mean angular dis- tance from perihelion. Adding a p to each member of Eq. (e), making m(t- t p ) -f a p = a w and writing a a p for F, we find a m = a 2 e sin (a a />) + f & sin 2 (a a p ) but, Analytical Mechanics, 192, dv _ f 7i-7' which substituted above gives, after making dr is Eq. (100) of the text APPENDIX VII. APPENDIX VII PLANETS' ELEMENTS. From Eq. (277), Analytical Mechanics, we have whence, making fx = k, 2c= f. vx 1 '); and this in the equation dv 2c Tt = ~7' Appendix VI., gives dt~ * and substituting the value of r* from the equation a (1 - f) I + e cos v ' we find y'- (1 + e cos v) f To integrate this, assume cos u e cos v = : 1 e cos u from which find the value of dv, eliminate dv and cos v above, and have, dt = = . (1 e cos u) du ; and by integration t + C = ( e sin ). But, Analytical Mechanics, 201, a* T Va ~~ 2*' in which 2" is the periodic time. 318 SPHERICAL ASTRONOMY. Whence, making t = when u = 0, #e have (7=0, and 2* ~ ./ = -esm; and denoting the mean motion by n, we lave 2* *= T ; and finally nt = M e sin ;. which is Eq. (106) of the text. The quantity t is the time from perihelion, for by malm g ti = 0, we have t = ; cos v = 1, or v = 0. APPENDIX VIII. PLANETS' ELEMENTS. Differentiate the equation and divide by *2rdt, we have dr x dx y dy z dz dt r ' d t r ' dt r'dt and making we have ;?' + * * + ; which is Eq. (112) of the text APPENDIX IX. 319 APPENDIX IX. PLANETS ELEMENTS. Make i, a*, a 3? the observed right ascensions; ^u & A: the observed north polar distances ; t>t ^ tai the mean times of observations reduced to any first meridian, say that of Greenwich ; and suppose the observed quantities corrected to the mean equinox and mean position of the equator at the beginning of the year. In the interval of time required for light to travel from a roaming body to the earth, the body describes some definite portion of its path, and at any given instant we see the place it left and not that which it actually occupies. We look, as it were, at luminous places on the orbit, but always behind the body's true place. The position which a body occupied at the instani the light started, and in which it is seen at a given time, is called its virtual place at that time ; and that which it actually occupies is called its true place. Conceive three sets of parallel rectangular co-ordinate axes, one set through the place of observation, another through the centre of the earth, and the third through the centre of the sun. Take the planes xy parallel to the plane of the equinoctial, the axes of x parallel to the line of the equinoxes and positive towards the first point of Aries. Denote by p y the distance of the body's virtual place from the earth at the time t t , and by v the time required for light to travel over the mean radius of the earth's orbit, which we have taken as unity ; then will v p ; be the time required for light to travel over the distance p,. Denote by a?, y, z the co-ordinates of the virtual, and .T, y, z the co-ordi- nates of the true place of the body at the time /,, referred to the centre of the earth ; then, regarding the motion of the body as uniform during the time v p,, will dx x = x vp,. ? = ,-.,. Yl dt dz 320 SPHERICAL ASTRONOMY. Denote the co-ordinates of the sun, cleared of aberration at the time / and referred to the same origin, by X t , Y,, Z { ; and the heliocentric co ordinates of the true place of the body at the same time by x t , y t , z, ; then will which in Eqs. (1) give x = X t + x, v P/ y = (Z. (2) in which p, = (s, + ZJ sec /3, (3) or, which may be preferable, if the body be near the equator, p y = (ar y + -X)) sec a, cosec /3, (4) Denoting the co-ordinates of the virtual place of the body at the time ,, referred to the place of observation, by ', y', z' ; and the co-ordinates at the same time of the place of observation, referred to the centre of the earth, by/,, g t , and h\ then will ~z =z' + A; which substituted in Eqs. (2) give But y' x' tan a, = z x' tan 0, =s (5) (6) APPENDIX IX. 321 in which cotan 0, = cos a t . tan /3, ....... (7) Also, if I denote the geocentric colatitude of the place of observation, p the corresponding radius of the earth, and T t the sidereal time of observation. reduced to degrees, then will f t = p . sin I . cos T t \ ^=,p .sin /.sin T , ( ....... (8) h = p . cos I . ) ind sun's horizontal parallax at the place of observation ' ~~ number of seconds in an arc equal in length to radius ' " . Multiplying the first of Eqs. (5) by tan a z and subtracting the product from the second, then by tan 0, and subtracting the product from the third, and reducing by the relations of Eqs. (6), we have in like manner y - and (10^ in which, as in equations (3) and (4), P3 = (z, + Z.) sec or, if the body be near the equator, p 2 = (a* + ^2) sec 2 . cosec p, = (x 3 + JT,) sec a, . cosec Now make <, = ^ 8 T, and Jj = t, + r f ; 21 ) > SPHERICAL ASTRONOMY. then, because ar, and # 3 are functions of t, which become 2, when r and become zero, we have by Taylor's formula, * l - x * ~ Tt ' c + and the same for y, and , dx, d*x z r' 8 _.._ _ r ._ _._ and the same for y 3 and 3 . "(13) The intervals r and r f must be such as to make these expressions con verge rapidly, and it will rarely if ever be necessary to retain the terms of the series involving powers of COS 6, COS d 3 r- r' / r'\-i ^ = (Jg- S) sin (*,-.) ' P- 7 + ^~ ^l 1 +7/J cos a, cos a 3 COS , cos (24) or by making cos a, cos a, - sn a- cos d, cos ^ (R - S) sin (0, - we have p = D (a, - a,) + ^ (a, - (25) and making we have 326 SPHERICAL ASTRONOMY. G (6 8 - 6,) ^ tan tan ~~ = tan a, - (x a tan a, + A { + a, y f = tan ^ - i (a, tan d t + ,+&,- ,) or instead of the last two, = - tan a, + (or, tan a 3 + A, + 3 - 2 = tan 3 + tan ^ + ^a + 6. - ' (26) Now although the Eqs. (26) express the values of the co-ordinates and components of the velocity of the body at the time of the second observa- tion, they involve the geocentric distances p,, p 2 , p l? and the radius vector TS, which are unknown, and the solution of the problem can only be ac- complished by successive approximations. First Approximation. Let us first neglect the terms involving aberration, and those containing as factors powers of , which with a, and bi in fourth, fifth, and sixth, or fourth, seventh, and eighth of (26), give -~, - , and -^ t and the values of p,, p 2 , pi in Eqs. (19) or (20). Second Approximation. By differentiating the equation r 2 8 = xf + yf + gf, and dividing by r 2 dt, we have ry _X 2 jK* y f y^ z a ^ dt ~ r, ' dt "*" r 2 ' dt ^ r t ' dt 330 SPHERICAL ASTRONOMY. The first terra of this equation becoming thus known, the values of 7, W, U r , and TF 7 , Eqs. (17), may be computed to include the third powers of r and 2 , Aa 3 , A& 3 the difference between the first and second values of the quantities written after the symbol A, *ftid observing a like notation for the other quantities, we have for computing d x 2 dy% d z% the Hrst corrections to # 2 , y 2 , z 2 , , : . and - from the third and at at at foui i of Eqs. (25), Ap = D (Aa 3 A 8 ) + E (Act, Aa 2 ) ) A q == F (A 6 8 - A & 2 ) + G (A 6 t A 6 2 ) ) . .(41) an den from Eqs. (26), A x s = Ap A q A y 8 = A x t tan a a + A a s A % = A # 8 tan 4 2 + A 6 2 A - = A 7~ 2 tan a, (A x tan a, -f- A a, A y 2 ) at at T ^ A = A 1 tan ^ - (A* 2 tan 6, + Ab { - A* 2 ) ' . . (42) Third Approximation. Differentiating equation (40), dividing by d t, and substituting for r^-> -TT -p, r^-> -TTi tn ^ r values in equations (14), we have d? df df dt> r (43) r with this value for -j-j , find new values for Z7", TF", IT, and IF 7 from Eqs. (17); and for a,, 6,, a 2 , & 2 , a, 3 , J 2 from Eqs. (18), by including the terms that were omitted before. Then with the differences between these last values and the next preceding, form equations for the final corrections by writing A 2 for A in equations (41) and (42). Then the final values of the required quantities become APPENDIX X. 331 &c. = ar, &c. = y, &c. = z; APPENDIX X. GEOCENTRIC MOTION. By the notation of the text, p. 91, a cos / cos Z = p . cos X, a sin / sin j& = p . sin X ; and by division, a sin I sin L tan X = - - -- - ; a cos / cos L differentiating, d A _ (a cos 2 cos L)(a cos 2 . dl cos L . dL) + (aaml sin Z)(a s'ml.dl sin LdL) cos 2 A (a cos ^ cos Z) a _ [a a cos (L 1}] dl+ [1 a cos (L I)] dL (a cos I cos Z} 8 But by Kepler's 3d law, dL : dl :: a^ : i; whence dL = a? .dl; which substituted above, and making p = a cos ^ cos L ' gives d X:= P 8 . [a 9 + a? _ ( a + J) cos (Z - /)] . rf/; and making d\ m, and dl = n y we have Eq. (124) of the text. 332 SPHERICAL ASTRONOMY ! APPENDIX XI. . ," | ON ECLIPSLS. BY MR. W. 8. B. WOOLHOU8E, HEAD ASSISTANT ON THE NAUTICAL ALMA VAC ESTABLISHMENT. Eclipses, in all the varieties of aspect which they present to different places on the earth, form an entertaining subject for discussion; and, without considering the public interest generally excited by their prediction and appearance, the use of them, as a test of the degree of perfection of the lunar and solar tables, and in the determination and corroboration of geographical position?, Moon's hourly motion in latitude I f reatest 3 4 least Inclination of moon's orbit with ecliptic . I f reatest 5 20 6 f least 4 57 22 LIMITS. For the occurrence of an eclipse of the moon : 1. The greatest possible distance of the centres of the moon and earth's shadow at the time of contact, is 63' 29". 2. At the time of true ecliptic conjunction of the moon and earth's shadow, or at the time of opposition or full moon, the greatest possible latitude of the moon is 63' 45". 3. At the time of opposition, or full moon, the greatest possible distance of the centre of the moon or of the earth's shadow from the ascending or descending node of the moon's orbit is 12 24'. For the occurrence of an eclipse of the sun : 1. The greatest possible distance of the centres of the sun and moon, at the time of contact, is 1 34' 28''. 2. At the time of true conjunction of the sun and moon, the greatest possible latitude of the moon is 1 34' 52". 3. At the time of true conjunction of the sun and moon, or the time of new moon, the greatest possible distance of the centre of the sun or moon from one of the nodes of the moon's orbit is 18 36'. The third of these limits applies to the true place of the node, which may differ considerably from the mean place. The most convenient and certain limits, however, will be those of the moon'f latitude (/?), and will be as follows : 1. At the time of full moon an eclipse of the moon will be certain ) , ., ( < 51' 57' 1 impossible P JD M >6345 and doubtful between these limits. 334: SPHERICAL ASTRONOMY. For the doubtful cases, an eclipse will result when in which P, s denote the equatorial horizontal parallax and semi-diameter of the moon, and T, o those of the sun. 2. At the time of new moon an eclipse of the sun will be certain { when/JJ <123'15' impossible > I > 1 34 52 and doubtful between these limits. For the doubtful cases, an eclipse will happen when & <(,?_- *)+,+** + 25' PARALLAX. If a straight line be drawn from the centre of the earth to any assumed place, it will be the radius of the earth for that place, and this radius we shall designate by the letter p. This radius p, produced upward towards the heavens, will determine what we shall call the central zenith, being that point which spherically deter- mines our true position in relation to the centre of the earth. The apparent ze- nith, however, is naturally determined by a line which is vertical to the observer, and therefore a normal to the spheroidal surface of the earth. The small angular deviation of this normal from the radius of the earth, or the angular distance be- tween the central and apparent zeniths, is what astronomers call " the angle ol the vertical ;" and, the earth being an oblate spheroid, it is evident that the cen- tral zenith will be nearer to the equator than the apparent, and also that the hor- izontal parallax will always be less than that at the equator, in consequence of the diminution of the earth's radius in proceeding towards the poles. The effect of parallax on the position of a body above the horizon is to augment its zenith dis- tance, and for this we have the well-known relation, " sin par. in zun. dist. = sin hor. par. X sin app. zen. dist." This relation will hold strictly for the spheroidal figure of the earth, provided we adopt the central zenith, and that horizontal parallax which appertains to the ra- dius p of the place of observation. Consider the equatorial semi-diameter of the earth as unity, and let y denote the polar semi-diameter, which, adopting the mean between La Lande and Delam 304 bre, will be - . Let also / be the latitude of the central zenith, or what is usu- 305 ally called the " geocentric latitude," and I that of the apparent zenith, which may be termed the spheroidal or geographical latitude. Then the co-ordinates of this place, referred, in the plane of its meridian, to the polar axis, will be x = p sin I, y = p cos I. By the generating ellipse and therefore for the angle T, which the normal makes with y or the tangent with , we have dy 1 x tan I tan I 1 = - =s . - = , dx Y' y y a .-. tan / = y tan I' . . . . APPENDIX XI. 335 Again, the values of x and y, substituted in the above equation of the ellipse, give and hence " ' To these may be added the following, which are sometimes useful, and directly deducible from the equations (1), (2), y* tan /' (1 - *') sin /' x = a sin / = = = , _ = .... (3) Vl-fy 2 tau u /' VI-** sin* I' 1 cos I' y = p cos I = - = .... (4) _ V 1 -f y' tan 2 *' VI - e* sin' f where /' - / = ( 2y 1 ~ n y i ,,) n 2 x, in seconds, will be equally adapted to both reductions, giving nearly the mean between them ; and a table so constructed, with the argument x signifying either latitude, will answer every necessary degree of accuracy, since the reduction itself is so small fn numbers we have -~^-' = , and its logarithm = 7.51641 .'.log ( ^^ ) =r 2.83084, 2y 2X304X306 6 \2ytanl"/ and hence I' - /= [2.83084] sin 2 X . 336 SPHERICAL ASTRONOMY. Thus the following table has been derived : Difference between the Geographical and Geocentric Latitudes. Argument: %, either Latitude. X V I X I' I i X I -I , o o i ,i o o , o 90 c c i5 7 5 5 3 9 3o 60 9 4 7 I 89 c 24 16 7 4 5 5 9 3i 59 9 58 2 88 o 47 17 7 3 6 19 32 58 10 9 3 87 i ii 18 72 6 38 33 5 7 10 19 4 86 i 34 19 71 6 5 7 34 56 10 28 5 85 i 58 20 70 7 i5 35 55 10 3 7 6 84 2 21 21 69 7 33 36 54 10 44 7 83 2 44 22 68 7 5i 3 7 53 10 5 1 8 82 3 7 23 67 8 7 38 52 10 57 9 81 3 29 24 66 8 23 3 9 5i n 3 10 80 3 5a 25 65 8 39 4o 5o ii 7 ii 79 4 i4 26 64 8 54 4 1 49 ii ii 12 78 4 36 27 63 9 8 42 48 ii 14 i3 77 4 5 7 28 62 9 22 43 4 7 ii 16 i4 76 5 18 29 61 9 34 44 46 ii 17 i5 7 5 5 3 9 3o 60 9 4 7 45 45 ii 17 The difference is to be subtracted from the geographical, or added to the geo- centric latitude, whether it be north or south. It is evident from what has been said, page 334, that if Z denote the true dis- tance of the moon from the central zenith as it would appear at the centre of the earth, and Z' the apparent distance from the same zenith, as seen from the place on the surface, where the radius of the earth is p ; and furthermore, P the equato- rial horizontal parallax, and z = Z' Z, the parallax in altitude, we shall have sin z=/ sin P sin Z' Substituting Z -f- z in the place of Z', and dividing by cos z, we find p sin P sin Z tan z = 1 p sin P cos Z which are the usual formulae for the parallax in altitude. For the radius p of tne earth we have log (8) (9) v a JL. = 8.909435, and by (6) tan $ = [8.909435] sin /, p = cos $. The values of p so computed are given in the annexed table. APPENDIX XI. 337 Log. Radius of the Earth. Argument: 'Geocentric Latitude. I % P I log f J log f o o O o .00000 3o 9.99964 60 9.99893 I o .00000 3i 9.99962 61 9.99891 a o ooooo 32 9-99960 62 9.99889 3 o . ooooo 33 9.99958 63 9.99887 4 9.99999 34 9-99955 64 9 . 99 885 5 9.99999 35 9-99953 65 9 . 99 883 6 9.99998 36 9-99951 66 9.99881 7 9.99998 3 7 9-99948 67 9-99879 8 9.99997 38 9-99946 68 9.99877 9 9.99997 3 9 9.99943 69 9.99876 10 9.99996 4o 9 .99 9 4r 70 9.99874 ii 9.99995 4i 9-99938 71 9.99872 12 9.99994 42 9-999^6 72 9.99871 i3 9.99993 43 9.999^4 73 9.99870 i4 9.99992 44 9.999^1 74 9.99868 i5 9.99990 45 9.99929 75 9.99867 16 9.99989 46 9.99926 76 9.99866 '7 9.99988 47 9.99924 77 9.99866 18 9.99986 48 9.99921 7 9.99864 '9 9-99985 4 9 9.99919 79 9.99863 20 9.99983 5o 9.99916 80 9.99862 21 9.99982 5i 9.99914 8r 9.99861 22 9.99980 52 9.99911 82 9.99860 23 9.99978 53 9.99909 83 9.99859 24 9.99976 54 9.99907 84 9.99859 25 9.99974 55 9.99904 85 ' 9-99858 26 9.99973 56 9.99902 86 9 . 99 858 27 9.9997! 5 7 9.99900 87 9.99858 28 9-99968 58 9.99897 88 9.99858 o 9 9-99966 5 9 9.99895 89 9.99857 3o 9.99964 60 9.99893 9 9.99857 PHENOMENA WHICH TAKE PLACE ON THE EARTH GENERALl Y. The place on the surface of the earth where the limbs of the sun and mocn first appear in contact will be where the penumbra first touches the earth, and, conse- quently, at this place the apparent contact will be in the horizon, the disk of the moon being wholly above the horizon, and that of the sun below it. The point of contact will l>e in the same vertical with the two centres; and, therefore, the real as well as the apparent places will be in the same vertical circle ; and the lower limb of the moon, being in the horizon, will be depressed by the whole amount of the horizontal parallax which belongs at that time to the latitude of the place. Similarly, the place which first has a cr ntral eclipse will be where the straight line through the centres of the sun and moon comes first in contact with the earth, and at this place the centres of both objects will be in the horizon, that of the moon experiencing the whole effect of the horizontal parallax. 22 338 SPHERICAL ASTRONOMY. The same circumstances will have place where the phenomena finally quit the earth. Since the apparent places of the sun and moon are so contiguous, and the par- allax of the sun so small, it is evident that the relative positions will be the same if we give to the moon the effect of the difference of the parallaxes P it, and retain the sun in his true position. This difference P it is therefore the relative parallax, or that which influences the relative position of the bodies. If p be the radius of the earth for the place on its surface, the parallax which ought to be used ie p (P it). But in the following investigations, where a place is generally the object of determination, we cannot previously so reduce this relative parallax P it. In order therefore to secure the chance of least deviation from the truth in this respect, we shall in these cases reduce the parallax in the first instance to a mean latitude of 45, so that it will be [9.99929] (P v). We shall conse- quently, to simplify the analytical expressions, hereafter denote this quantity by the letter P' only ; except in one or two instances, where the latitude of the place is known, and where it is always distinctly specified to represent the parallax properly reduced to that latitude, or p (P *). I PLACES WHERE THE DIFFERENT PHASES ARE FIRST AND LAST SEEN ON THE EARTH. Let the whole be referred to the sur- . Fig. 5. face of a sphere concentric with the earth ; and let R be the relative orbit of the moon, which is generated by the differences of the motions in right as- cension and declination, or by the rela- tive motion of the moon ; N ihe north pole ; S the sun ; Sn perpendicular to the relative orbit, the nearest approach v ^ which we denote by n\ G the point ^T where the rnoon comes in conjunction in right ascension, and OS the difference of declination at that time, which we denote by contraction, diff. dec. Let also MM' be the positions of the moon, when a distance of the centres equal to A' first appears on, and finally quits the eaith ; M S=M' S= A, the corresponding true distance as seen from the centre of tho earth ; ZZ' the zeniths of these places on the earth, which must be respectively in the continuations of SM, 8 M 1 , in order that the full effect of parallax may be communicated in causing the bodies to approach. As the apparent zenith distance of the points which experience the greatest effect must be 90, we may evidently assume Z$ = 90: for contact of either limb of the moon with the contiguous limb of the sun, we have accurately Z/S = (90 TT) -f- a ; for contact of either limb of the moon with the remote limb of the sun ZS=(90 -*) ; and for contact of the centres Z=90 *. By making Z $ = 90, the phase will begin with sunrise and end with sunset ; and it is evident that no sensible augmentation can affect the semi-diameter of the moon so near the horizon. The true distance S M of the centres being A, and f* the relative horizontal parallax, the apparent distance A' will be P' ~ A ; and by estimating positive distances from S towards M, in order to have the first oc- currence of the phase, it will be A P' ; .-. A=P'-f A'. APPENDIX XT. Here we may notice three limiting aspects, (1) When simple or exterior contact of limbs first takes place, A' = * -f ff, and A = P r + s -f- a- (2) When interior contact of limbs first takes place A' = s ~ ff ; when * > r. a total contact first commences with A ' = a ; when s < o, an annular con tact first commences with A ' = a s. Therefore, If s > <7, a total eclipse first begins on the earth, when If s < , the angle under the distance and the line of nearest approach, or the angle MS n. This angle is always measured on the northern side of the dis- tance, so that when R falls below S, or when diff. dec. C 8 is negative, it will exceed 90. Then the relations of the figure will give these equations : tant = - 1 =; n = (diff. dec.) cos < ...... (1) ii cos D Di Hourly motion in the orbit = - , sin i arc n G = n tan t. For the time of describing the arc n (7, or the interval between the middle of the general eclipse and the time of conjunction, it must be divided by the hourly motion in the orbit. Therefore, t denoting this interval, tan c. Assume (n sin t ~BT . = 3600" X = [8.66630] ..... < 2 > t in seconds = c tan i J The sign of will be determined by combining the signs of diff dec. and D\ ; and then time of middle = time of <5 t ........ (3) Also cos = ........... (4) A Mn = n tan w . 310 SPHERICAL ASTRONOMY. Let T denote the semi-duration of the phase, or the time of describing Mn, and T in seconds = c tan i u~. Time < Again, let, at the beginning, the / NSZ = a, and for the ending, the / N S Z' = 6; and, these angles being estimated from N8 towards the east, we Bhall have = (-,)_, & = (-,) + (6) and, the sun being supposed in the horizon, Z S = 90, Z' 8= 90, cos NZ =cosNSZ sinNS, tanZNS=- ^^r, cos NS cos NZ' = cosN8Z' sin NS, tanZ'NS = - i&nNSZ '. cos N8 tan a em I = cos a cos 6 ; tan h = sin <5 tan 6 sin /' = cos b cos 5 ; tan h' = : - sin S the latitude and hour angle /, h, relating to the first place, and /', h', to the last. These hour angles are measured from the sun towards the east, so that the longi- tudes of the places will be determined by subtracting respectively from them the apparent Greenwich times of beginning and ending reduced into degrees and min- utes, observing that positive differences will indicate east longitudes and negative differences west longitudes. In the preceding formulas we must use, f Partial ~] f P' -f s + Mm - Sm, or A < P f -f A' and > P' A', that is, A must be between the values P' A' and P'-f- A' : this leads to two spe- cies of curves. 1. When the nearest approach is greater than P' A '. Here the formation of the triangles Sm M, 8m' M, will always be possible du- ring the appearance of the phase on the earth. At the first appearance and final departure of the phase, 8 M = Mm -f- Sm, the triangle Sm M will be simply the line S M, and only one place Z will result. By taking positions of M on both sides of the middle point n, it will also appear that the relative positions of the places Z Z f become inverted, and that the curves described by them must intersect each other at some intermediate place. Hence it appears that the curve of risings and settings commences with a single point, which immediately after divides itself into two points moving in opposite directions on the earth, and which describe two curves intersecting each other, and finally meeting again in a single point, the whole forming one continued curve, returning into itself, and assuming the figure of an 8 much distorted. At the place where they intersect, the phase will begin at sun rise and end at sunset, or it will begin at sunset and end at sunrise. 2. When th,e nearest approach is less than P' A '. In this case the triangles SmM, Sm' M, will resolve into the line $ Jtf" when A = P' + A ' and also when A = P' A ', each of which positions will give only one place Z, Thus it appears that the points Z will form two distinct, oval, and isolated curves, the former curve being generated between the decreasing values A =P / -f- A' and A = P' A'-, and the latter between the increasing values A = P' A 'and A = P' -f- A'. The leading point of the first oval and the terminating point of the second oval are the places where the phase begins and ends on the earth. The terminating point of the first oval and the leading point of the second oval are simply determined by using A = P' A', and computing the same as for the beginning and ending of a phase on the earth. Let us now turn our attention to the determination of the two places Z Z', at any time, or for any position of M. Join Z S and draw Md perpendicular to NS. We shall, throughout our investigation, usually denote Sdby (x), d M by (y\ and the / dS Mby S, this angle being estimated from S JV towards the east. To determine these quantities, let the declination of the point d=(D), which will a little exceed that of M, and which is distinguished from it by being placed within a parenthesis; then, supposing N M to be joined, the right-angled spherical triangle Nd M will give tan (D) = . As a is always small, the difference of the declinations (D) D-= tan- 1 D maybe arranged in a small table M annexed 342 SPHERICAL ASTRONOMY. Difference between (D] and D, or a corr. Arguments : D and a. a D 10 20 3o 4o 5o 60 70 80 00 IOO o ,, II O O o o 3 o o o O o I o O o i I i I 2 2 o o. o i i I 2 2 3 3 o I i 2 2 3 4 5 4 o o I 2 2 3 4 5 6 5 j 2 3 4 5 6 8 6 o o I 2 3 4 6 7 9 7 o 2 3 4 5 7 9 ii 8 o o 2 3 4 6 8 10 12 9 o 2 3 5 7 9 ii i3 10 o 2 4 5 7 10 12 i5 ii c 3 4 6 8 10 i3 16 12 2 3 4 6 9 ii i4 18 i i3 o 2 3 5 7 9 12 i5 19 i i4 o 2 3 5 7 10 i3 17 20 ! t5 o 2 3 5 8 ii 14 18 22 16 o 2 4 6 8 ii i5 I 9 23 17 2 4 6 9 12 16 20 24 18 o 2 4 6 9 1 3 16 21 26 1 9 o 2 4 7 10 i3 17 22 2 7 20 ' 3 4 7 TO i4 18 23 28 21 o 3 5 7 TI i4 IQ 24 29 22 o 3 5 8 II i5 IQ 25 3o 23 o 3 5 8 II i5 20 25 3i 24 3 5 8 12 16 21 26 32 25 o 3 5 8 12 16 21 27 33 26 o 3 6 9 12 17 22 28 34 27 3 6 9 13 17 23 29 35 28 o 3 6 9 13 18 23 29 36 29 o 3 6 9 i3 18 24 3o 3 7 The number of seconds given by this table, which we have denoted by the term a corr., is to be applied so as to increase D, whether it be north or south. The value of (Z>) being found by so correcting D with this table, we shall evi- dently have --8 __ cos 8' (A) the quadrant in which S is to be taken being determined by (x) and (y) as co- ordinates. APPENDIX XI. 343 We shall afterwards have frequent occasion to use these quantities. If t denote the time from the middle of the general eclipse, they may be deter- mined more easily, though less accurately, by means of the following formulae, which may readily be inferred from what has preceded. t tan w = -, A = (B) e cos w ' 8--= (-4*0, (x) = A cos S, (y) = A sin 8, . the upper sign being for the time t before the middle, and the under sign for the same time after the middle. Denote the / mMS by m. In the triangle mMS, which may, on account of its smallness, be considered as a plane one, we also have Jf/w=P', Sm= A'. and S Jt/ = A. Assume _P' A' and then P' . A As ZS,Zm maybe considered as quadrantal arcs, they will be parallel at the extremities S, m; and thus the / ZSM~ /. mMS~m. Therefore the Z N 8 Z S m ; and the sun being supposed in the horizon, the spherical tri- angle NSZ will have ZS=9V, and hence the places Z, Z', will depend on the following formulae, in which Z is called the place advancing, and Z' the place fol lowing. Place following, sin I = cos (S m) cos J. tan h = * . "T , sin 3 , . . Place advancing, | ' tan (8 4- m) em I = cos (S + m) cos <5, tan A = V i , sin o In these expressions the symbol S represents the declination of the sun at the time for which we calculate ; but for common purposes the value of S at the time of conjunction may be used in all cases. III. NORTHERN AND SOUTHERN LIMITS FOR ANY PHASE. The determination of the extreme latitudinal limits of a phase, or of the terres trial lines whereon that phase will appear as the middle of the local eclipse, is the most complex and unmanageable of all operations which relate to a general eclipse. For any given phase, at different places on the earth, the moon must be so reduced by parallax as to touc'i a given concentric circle on the solar disk ; and if we con sider this circle, by way of illustration, to represent, instead of the sun, the disk of the luminous body, the places on the earth which severally see the given phase must be situated in the surface of the penumbral or umbral cone, according as the interfering limb of the moon only approaches or projects over the centre of the sun; that is, the places must all be found in the intersection of this cone with the surface of the earth. This intersection will assume a complete or partial oval ojffiERICAL ASTRONOMY. lorm, according as the cone falls wholly or partially on the earth's illuminated disk. When it falls only partially on the earth, the extreme points will evidently see the sun in the horizon, and be therefore two points belonging to the horizon limits ; -but in the other case the phase cannot at that instant be seen in the hori- zon. It is evident then, that these two cases have been already characterized in the discussion of the rising and setting limits. Let us now suppose the bodies to assume consecutive positions, answering to very small intervals of time, the earth also turning round its axis, and we shall have a series of these ovals. It is obvious that the extreme geographical limits of the phase will be represented by curves which envelope all these ovals; that at each instant the place of limit, by reason of the compound of the motions, will be proceeding relatively in the direction <>J the tangent to the oval ; that there will be two of these limits when the oval becomes entire during the eclipse, but only one when it is always partial. This is the most popular and natural idea that can be formed of the nature of these limits ; and we may here remark, as an inference from what has been said, that if the rising and setting limits of any phase do not extend throughout the general partial eclipse, there will be both a northern and southern limit to that phase ; but that, on the contrary, when the rising and setting limits continue throughout the eclipse, there will be only one of these limits to the phase, viz. : a southern limit when the difference of declination at conjunction is positive, and- a northern one when that difference is negative. As before, let the system be referred to a sphere concentric with the earth, and let M be the place of the moon ; Z, Z', the zeniths of the places which are respectively in the northern and southern limits ; and m, m', the corres- ponding apparent places of the moon. Draw the meridians N m', NS, N m, N Z, NZ',; aiso m r, m' r', and M h d h' perpendicular to ATS; and assume Sd=(x),dM=(y),m h = x, /_ N rn ZM, / m N S = a', declination of m = D', and the latitude of Z=l. Then the / mNZh a\ m MZ 1 Bin Z, x = mMco& MP' sin Z cos M and y = m M sin M = P' sin Z sin M ; these by spherics resolve thus : x = P' sin Z cos M = P 1 [sin / cos D' ->- cos / sin D f cos (h a') ] y = P' sin Z sin M = P' cos / sin (A a') From these we deduce =(*) P' sin Z cos M (x) = P' [sin / cos D' cos / sin ' cos (h a')] (x) , = (y) P' cos I sin (h a ') Lt us now keep our attention to the same place Z on the earth, and suppose the system to be in motion as in nature. The hour angle h will increase at th APPENDIX XI. 34-5 rate of 15 per hour, and the latitude / will by hypothesis remain unchanged; *t that the following equations will ensue : - = P 1 sin 1" -j- [sin I sin D' -f cos I cos D' cos (h a ') ] d t at + P' sin l'Yl5 ^\ cos / sin D' sin (h ') ^ V a t / at = P' sin 1" d -^- cos Z+ P sin I"(l5 ~) sin D' sinZsin Jf ^ at \ or / dt !L = ^} _ P ' 8ia 1" (150 - %) cos / cos (A - a') a a \ at/ = 1^) __ p' 8 in 1" (i 6 ^-\ (cos Z cos ' sin Z sin D' cos M ). at \ at/ Now, in order that m may be the apparent place of the moon at the middle of the eclipse, and consequently her nearest apparent contiguity with the sun, we must have = ; or since u*-{-v* A /2 , u (- v = 0, which is the condition of a t at at limit . Before we substitute the preceding values of - , , it may be observed, to at dt avoid complexity, that the quantities P 1 sin 1'' , P 1 sin 1" - may be neg- d t d t lcted as being very small compared with P'. 15 sin 1", ~ ' and -^ ; also that & may be substituted for D', which will equally serve the purpose of both northern and southern limits. With these modifications we have ~ = P'. 15 sin 1" sin S sin Z sin M ^ = _ p, 15 o 8in !/, ( C08 ^ cos 5 sin Z sin i cos M ) at dt and, for the condition of limit, u [p r . 15 sin 1" sin J sin Z sin M - + v l^~ P'. 15 sin 1" (cos / cos i sin Z sin S cos JIf )1 = 0. Instead of P 1 sin Z cos M put (a;) -f- w. and for .P' sin Z sin Jf put (y) v, and it becomes Fl5 sin 1" (y) sin t J + v Fl5 sin 1" (*) sin t + ^ P' v 16 sin 1" cos Z cos 6 = ; . *. cos Z 346 SPHERICAL ASTRONOMY. But, if ai denote the true relative motion in right ascension, and Di the true relative motion in declination, and D the declination of the moon, at the time ol true conjunction, .-. cosZ = V COS C Make now the following assumptions : ii cos D (O) (3) P 1 cos 5 in which (A), () may be used as constant quantities throughout the eclipse, and we get cos Z = ( u sin v -j- v cos v). v The angle r S m is equal to the inclination of the apparent relative orbit with the parallel of declination; denote it by t', and then u = A' cos t', v = A' sin i', and (4) which is a concise form of the condition to be fulfilled by Z and t', in order that the place Z may be situated in the limit of a phase. Since the / MS d S, and the Z M S m = 180 (8 + *'), /. MSm' S~{- i, we have for the triangle MSm Mm* = A 2 + A' 2 2 A A' cos (8 + ') Divide this by P' 8 and we get for the geometrical relation between S and t', the upper sign applying to the northern, and the under sign to the southern limit. Add this to the square of the preceding equation (4), and there results for the determination of the angle t'. The solution of this equation is by no means very practicable ; but as a small error in the value of Z will not sensibly affect the angle i', we may have recourse to the following indirect process, in which we first consider the angle i' to be equal to i, which in most instances is very nearly so. The letter M designates the the angle Mm h. APPENDIX XI. 347 M = A SOS I v = ' sin t tttJf (D) = D + (a a') corr. (a a') COS (D) Z = (Z>) - D' = *_ __. y P' sin Jf P' cos Jf the upper signs being for the northern, and the under signs for the southern limit Or, if t be the time from the middle of the general eclipse, and w ' the angle under Mm and the line of nearest approach, we shall have Mm sin u ' = n tan u> = n , and Mm cos ' = n A', which, observing that Mm = P' sin Z, give the following equations, wherein E and F are constant for all the computations. n A e(n A') tan a/ = northern southern F I limi limit. cos *>=( (8) before ) ,, after f th The sign of the constants JE, F, are the same as that of n A' ; and when this is negative, the angle w' will be in the second quadrant. The value of Z determined in this manner will be sufficiently approximate for the purposes of a general map; and where greater minuteness is wanted, it will serve very well to get the angle i from the equation (4). For this we have COB Z cot *' = cot v A sin v which may be resolved thus : tan 2* cosv tan '=^^ (9 > After t' is so found, which is only wanted roughly, the accuracy of the calculation may be tested by the equation (4) ; and then we may proceed to a correct compu- tation of M Z, by the equations (7), only using t' instead of i. We shall thus have in the sphorical triangle Zm N, ZM=Z, Nm = 90 D', and the angle Zm N=M; and I V spherics the following formulae: tan 9 = tan Z cos M tan (h a') = cos(0 + ^) ta check sin tan j = tan sin Z cos M CO8 ft (10 cos (e -f D'} cos (h a') cos I For a map the equations (8) and (10) will alone be amply sufficient. In fact, where a very accurate calculation is wanted, the most satisfactory method will consist in first computing the places roughly ; then to reduce the horizontal paral- lax to the latitude by means of the radius p, from the table at page 337, and with 34:8 SPHERICAL ASTRONOMY. the use of the value of Z, to find the augmented semUdiameter of the moon by means of the table at page 360, and thence the proper value of A ', and then to follow the equations (3), (9), (4), (7), (10). The first and last points of these limits will have Z= 90. For these places we have therefore by (5) P"= A 2 + A' 2 '2 A A'cos(S+i'). If we assume i = t, we shall obviously have -8 + *' = $+=&>, and A cos (8 + i) =n, w being the angle under the distance A and the nearest ap- proach n, as before used. .'. P' a = A 3 + A' 2 2 A'n Consequently A 2 sin 2 = A 2 w 2 = P'3 ( n A') 2 , which divided by A 2 cos 2 o = w 2 , gives tan = - V P' 2 (n A ')*. Therefore by taking the constant c used in the computation of the beginning and ending of a phase on the earth, we shall have semi-duration = c tan <,, = v P /z (w A ') a , M which may be arranged for calculation as follows : n A' P' . , -} cos w = , semi-duration = c sin w , j Time of | ^e^ture [ = iime f middle | + } 8emi - duration > The places of entrance and departure of the limits, by continuing the assump- tion ' = , may be hence calculated as for the beginning and ending of a phase only using 8 ^ u instead of S, thus : & ^ u = D' t For place of entrance, tan a I /io sin I = cos a cos D', tan h =. : =-7, r \* sm D f For place of departure, sin /= cos b cos D', tan A = sin.0" Having assumed t' = , the times and places so computed will only be approxi- mate, though sufficiently near for general purposes. For an accurate calculation, we must first determine the true value of i r . Since Z=90, the equations (9) give t' = v, which is also shown by (4). We may, therefore, with the quantities taken out for the respective times of entrance and departure, proceed with the equations (C), (3), use v instead of t in (7), and then the final results will be deter- mined by (10). It ought, however, to be observed, that it will be advisable to take the time of entrance in excess to the next higher integral minute, and to re- ject fractions of a minute in the time of departure ; since by fixing on a time a trifle without the actual limits, the value of sin Z would come out greater than APPENDIX X 349 unity, and the calculation rendered useless in consequence. The places so compu- ted will be accurately situated in the limiting lines, and though not strictly the first and last points of these lines, they will be very nearly so. IV. DETERMINATION OF THE PLACE WHERE A GIVES PHASE WILL APPEAR BOTH AI SUNRISE AND SUNSET. We have seen (page 341) that when the rising and setting lines of a phase ex- tend throughout the eclipse, they will compose the figure of an 8 much distorted. The point of intersection or nodus is a place where the phase will be seen to begin and end in the horizon ; that is, it will either commence at sunrise and end at sunset, or commence at sunset and end at sunrise. At the time of the middle of the eclipse, the sun will therefore be very nearly on the meridian : if diff. dec, and S are of the same sign, it will be midnight, because the pole of the earth will have the zenith and sun on opposite sides of it ; but when those values are of different signs, it will be noon at the place, for then the zenith and sun will be both on the same side of the pole. If r denote the semi-duration of the eclipse, which begins and ends with the given phase, r will express the semi-diurnal arc of the sun; and.-, tan I tan 5 = cos ( T jj) = cos (r . 15), which being nearly unity, we must have f ~ X or Z nearly = 90. Consequently for the values c dtt d v U * V ' d7' d~t' at the t " ne f the micl( * le of the ecli P se which will be either noon or midnight, we may assume sin Z = unity, and Jf=0 or 180. So we get, from the equations (1) and (2), page 344-5, = - (*) P', f = (y), Let ft denote the hourly motion on the apparent relative orbit, and i' the incli nation with a parallel of declination ; then , d v . , du ~ It' * 81 ~~ ~dt ' or, HBmi'=7)i ) , n cos ' = a, cos D [9.41796] P f sin i \ The condition for the greatest phase is u - \- v = 0, or u sin ' v cos '= that is, [ (x) P 1 ] sin ,' (y) cos <' = 0. If t denote the interval past the time of the true conjunction, we shall have (ar) = diff. dec. -f t A and (y) = ta l cos&; .: [ diff. dec. P'] sin i' t [Di sin ' -f a! cos D cos '] =s ; / A \ / -Di \ or, since Di = I I sin i, ; cos D = ( -^ I cos i, \ sm i / \ sm / [diff. dec. P'J sin ( ' t -^- cos (' ~ t) =a 350 SPHERICAL ASTRONOMY. Assume ^- _ COS (t ~ t) k sin t' sin t and then t = - K - , or since A = /* sin t ', D\ t = ^!l n _L or tin seconds = [3.55630] ^-1 ..... (3) When diff. dec. is negative, M = 180, and the lower sign of P' must be used ; or, as a general rule, P' must be used with the same sign as that of diff. dec., and, since I nearly = 90 ~ 5, we can previously correct the horizontal parallax for the place by reducing it to a latitude equal to the complement of J. The value of t be- ing found, we shall have at the place when diff. dec. and t> have j ^s*** \ signs, app, time of true 6 = j ^ [ < ( 4 ) which compared with the Greenwich apparent time of the true conjunction will show the longitude of the place. For the values of u and v we have u = ( diff. dec. P 1 ) t DI = k cos (t' ~ t) k sin i' sin t = k cos t' cos t, v = t at cos D = t DI cot i = k sin t' cos <. Let ri be the nearest apparent approach of the centres ; and the semi-duration r will be determined by the equations v ri A' sin u> sin i' n = r , cos o> = ;, T = = . Bin ' A A and thence the latitude by the equation, tan 6 Or, using the above value of v, k cos t A' sin . cos (T . 15) ' r = ~V~' tan ' = n ' (6) the latitude being of the same name as diff. dec. The middle of the eclipse will not have the sun in the horizon, except k cos t = A , T = 0, / = 90 ~ , and therefore, unless these particular values should happen, the place will not range exactly in the line whereon the middle of the eclipse ia seen at sunrise or sunset ; this line, which we are about to notice, will pass the intersection at a higher latitude, and will form a very small triangle with the rising and setting limits. V PLACES WHICH WILL HAVE THE MIDDLE OF THE ECLIPSE WITH THE SUN IN THE HORIZON. In the first place, we shall suppose the inclination of the apparent orbit to be the same as that of the true. The condition for the middle of the eclipse will then be simply to have the apparent place of the moon somewhere on the line of near- est approach. On both sides of S take Sm=Sm' =s + or, andm, m' will be the limits be- tween which the apparent place must be, in order that an eclipse may result. On the orbit make M' m' = P'. Then if iri falls between 8 and n, this will be the first position in which the ecliose can take place. But, if m' falls beyond the APPENDIX XI. 351 point ,, the first position of the moon F 'C- 8 - will be at M t where Mn P' ; and in this case, for each position between M and M' there will evidently be a position of m' on both sides of the or- bit, and consequently two correspond- ing places on the earth; when the moon arrives at M' the remote point m' will be receding from S, and will at that time get beyond the limit of an eclipse, so that the other point m' only will pro- duce an eclipse under the assigned conditions. Again, when m n is greater than P', it is evident that these limits will continue throughout the whole duration of M'M' or MM. When m n is less than P' t by making m M" = P' the limits for an eclipse will end at the point M", and it will be impossible throughout the duration of M" M". These two cases are the same as those distinguished in the rising and setting limits, page 340, s + a being the value of A '. To determine the times between which these phases are possible, or the semi- durations answering to the positions M, M 1 , M", we shall in each instance denote the angle M m n by the character o>, and the following equations will be readily deduced. (1) When n < P 1 (s + ff ), n + . . . . (1) w 2 > 90 when diff. dec. is negative. These semi-durations will give two times of beginning and ending; the one an- swering to the point M and the other to the point M". The middle of an eclipse in the horizon will take place from the first beginning to the second beginning, and from the second ending to the first ending. The places will be determined by producing m M to a distance of 90 from m. If a great circle be drawn through S, so as to be at this point parallel to m M, it will evidently intersect the former at a distance of 90 and determine the same place. We shall therefore, in supposing the places to be determined in this man- ner, have the following formulae : First place of beginning, wj = 90, sin I = sin cos 6, tan h = cot t sin S It must be taken in the 2d semicircle, or between and 180 , First place of ending, Change the name of the latitude of the place of beginning, and to the hour angle h apply 180. The results will determine the place of ending. Second place of beginning, tan a sin <5 sin I = cos a cos $, Second place of ending, sin / =r cos b cos t, h = tan h = tan b sin 6 . (8) 352 SPHERICAL ASTRONOMY. The second places of beginning and ending will be two of the extreme points of the lines traced on the earth. The other two extremes may be determined by computing cos w = - -^ - , and proceeding as before, observing that n must be considered positive, and > 90 when diif. dec. is positive. These four extiviut points are the same as those of the northern and southern limits, the phase being simply external contact. 2) When n > f (s + ff) and < s + ff, The places will be determinable throughout the whole of the first ^ . . (4) duration found as above. (3) When n > s+ ff , n (s + ff) cos w = - ^ - Z, Jr n must here be considered a positive quantity, and w will be > 90 C when diff. dec. is negative. The phase will continue throughout the whole duration, and the ex treme places may be computed from this value of w according . (5) to the equations (3). Having found the limits between which the phase is possible, the places for any intermediate times may be determined thus, t denoting the time from the middle, (D > 90 when diff. dec. is negative, and the places by the equations (3). If n < s -f- ff, suppose n to be positive, and compute = ( ) sin u. Then for times, without the limits of this duration, we may determine four places ; two with w < 90 and two with o> > 90, which will all fulfil the necessary conditions. The preceding results have been derived on the assumption of t' = t. They will be sufficiently approximate for a general drawing of the lines on a map, and more particularly as these phenomena cannot be subject to minute observation. When, however, from local circumstances or otherwise, greater accuracy is wanted, we must use the proper value of t' and the relative horizontal parallax reduced to the latitude thus determined. Since Z = 90, the condition for the middle of the eclipse, according to the equation (4) page 346, is ' v = or i' = v. Let the figure at page 344 represent the positions which answer to the particulars of the present case. Then as M m = Mm' = P', the Z Mmm' = / Mm' m. Denote this angle by 6; the angles Nm M, Nm M by M, M 1 ; and we shall have M=ov, 180 S , /. 8+v 9, SMm! =180 (8 + v + 0). APPENDIX XI. 353 With the triangles MSm, M Sm', we hence find sin = -p sin (S + v) ; pi 8in O 8 + y g ) <; sin (S + v) wlack, for computation, may be thus arranged sin (8 + v) __ sin (8 + v + 9) 9= -pi -i sin = $r. A; J to be + or but less than 90 ; Sm' = 9 , _ sin (S + v + 0) . (6) The points m, m', may in some cases be both on the same side of S, and the value of Srn is only necessary to indicate whether any portion of the sun id eclipsed or not. To have an eclipse, Sm, taken as a positive quantity, must be less than s -f ) cos h cos I (*) In the course of the general central eclipse, one of the places on the earth will have the central eclipse at noon. At this instant the bodies will obviously have 23 SPHERICAL ASTRONOMY. true as well as apparent conjunction in right ascension, and .*. A = diff. dec. and S = 0. This place is hence determined thus : _, diff. dec. 11/7 6inZ = , 1 = 6 + Z, (3) Z to have the same sign as diff. dec. App. time of true 6 = west long, of place, These equations (1), (2), (3), involve the horizontal parallax P', answering to a mean latitude of 45, which will be sufficiently near for ordinary purposes. Where an accurate result is wanted, the calculation must be repeated with the use of the equatorial relative parallax properly reduced to the latitude thus determined. The first and last places on the earth which see a central eclipse, are to be found by the formulae at pages 338-40. The preceding discussions comprise all that is necessary for the calculation of the lines which are shown in the ma-ps now inserted in the Nautical Almanac, and which are quite sufficient to indicate the general character of the eclipse that may be expected for any particular place. We might now proceed to show the appli- cation of these equations in the resolution of innumerable other curious and in- teresting problems; but such a field of speculation would not conform with the object of this paper, and may the more willingly be abandoned on the considera- tion that the means of solution may, in most cases, be readily elicited from the equations already established. The following classification of these equations will be found to exhibit, in a comprehensive form, all that will be requisite to direct and facilitate the operations of the calculator, and relieve the mind from any un- necessary reference or consideration. NOTATION. D = the D 's true declination ; d = the O's true declination; a = the true difference of right ascension in are, or ]) '& right ascension O's right ascension ; D 1 = the D 's relative motion in declination, or J) 's motion in declination 's motion in declination, ai = the D 's relative motion in right ascension, or the motion of the D that of the ; Diff. dec. = the true difference of declination at c5 in right ascension, viz., D 's declination 's declination, at that time; P = the D 's equatorial horizontal parallax ; TT = the 's equatorial horizontal parallax; P' = [9.99929] (P-); s = the D 's true semi-diameter ; = the 's true semi-diameter; A = the true distance of the centres ; Z>', a', s', A', the apparent values of J), a, s, A; = the angle under A and n: in all cases this angle is to be tak?n pos- itively, and between and 180. APPENDIX XI. 355 I. BEGINNING AND ENDING OF A PHASE ON THE EARTH. 1. (D, A and ai at 6); tan t = - ; n = diff. dec. X cos t ; ai cos D i of the same sign as DI ; n of the same sign as diff. dec. n sin t [3.556301 c = -- - - J ; sin i to be found by combining the preceding values of cos t and tan i; sign of t to be determined by diff. dec. X -Di. Time of middle = time of <3 t ; C partial "j T-, I central For 4- }- eclipse, A = I total \ annular J n cos w = ; T = c tan w. =(-)-; ft.= (_0 + c* 4. Place of beginning, (S at <5 ) ; . . tan a sm / = cos a cos S : tan h = : r ; sin i H= apparent Greenwich time of beginning; longitude east = h H\ h to be in the same semicircle with a. 5. Place of ending, (S at c5 ) ; sin / = cos 6 cos & tan h = : - : Bint ' JET= apparent Greenwich time of ending; longitude east = // H\ h to be in the same semicircle with 6. 6. For more accurate calculations, reduce the true relative horizontal parallax, oy means of the table at p. 387, to the latitudes so determined, and recompute. H. KISING AND SETTING LINES. For partial eclipse, A ' = -f- r. 7. When n > P' A '. These limits will extend throughout the entire duration of the general eclipse, and form the distorted figure of an 8, the first and last points being the places ol beginning and ending on the earth. 350 SPHERICAL ASTRONOMY. 8. When n < P - A '. "With P' A ', instead of A ', compute as for the times of beginning at 1 ending on the earth ; and let these times be t\ t fa. Then the risings j ^ in j at j P artial be g innin S> in which interval the first oval will be completed : in which interval the second oval will be completed. The limiting places at the times t\, fa, are to be found in the same manner aa the places of beginning and ending of a phase on the earth. 9. Places for any times within the limits : Prepare the constants, p - - - , q = - , and let t be the time from the middle of the general eclipse ; t n tan w = : A = - ; c cos m w > 90 when n is . 10. 8=(-t). the " A A ~YT ZT p ) f n 1 ,2 p ) V 2/ . m sin = , 2 r JT . A 771 to be less than 90 and positive 12. Place following, sin / = cos (8 m) cos S ; tan h = -. ; '; sin t H= apparent Greenwich time; longitude east = h H ; h to be in the same semicircle with 8 m. 13. Place advancing, tan (S -4- m) sin / = cos (8 -|- wi) cos ^ ; tan h = ^~"A ' ' longitude east = h H; h to be in the same semicircle with 8 + m. 14. For a more accurate determination, find the values of D, t, a for the given time, a*d P' = p (P w) for the latitude ; thence (D) = J) + (a corr. from table, p. 342) ; APPENDIX XI. 357 . wt sin * The quadrant of $ to be determined by (z), (y), as co-ordinates. With these values of S, m, compute the places by Nos. 12 and 18. When n > P' A '. 15 Find P' =p (P ir). for a latitude equal to the complement of $ at rf. /.sin*' = .01, n cos t' =i cos D [9.41796] P' sin S, . t . H mmds = COS (*' ~ ) 1 6. At the place, When diff. dec. and t have j J^^ 6 | signs, PP- time of true 6 = j ^h [ - *, which, compared with the Greenwich apparent time of the true <3, will determine khe longitude of the place. 17. k cos i A' sin < cos = -T-, T = , I to be of the same name as diff. dec. IV. PLACES WHERE THE MIDDLE OF THE ECLIPSE IS SEEN WITH THE SUN IN THE HORIZON. 1 8. When n < P' ( -J- ), compute c P' n(s + 2, u 3 , proceed as for the beginning and ending ot h phase on the earth. When diff dec. is +, j ^ a j. gives points meeting j l [ limit. When diff. dec. is -, j [ gives points meeting j ~^ | limit. The eclipse will be visible on both sides of the equator. 19. When n > P' ( + ff ) an * -\- a, compute 3, r s, as above. The phenomenon will continue throughout the whole duration, and the extreme places will be determined by proceeding with this value of w as for the beginning and ending of a phase. These places will in this case be also those ol first and last appearance. 21. Places for any time within the limits : Let t be the time from the middle, and compute If n < s + 8, this u may be taken both greater and less than 90 when t is greater than r 3 before found ; and then four places will be determined. In all other cases whatever w must be > 90 when diff. dec. is negative. The places to be determined by proceeding with w as for the beginning and end- ing of a phase. 22. For a more accurate determination at any time : Find P' = f (P IT) for the latitude before found. Find (a?), (y), 8, and A, as in No. 14. For the time of (5 form the constants (A) = [0.58204] a, cos D, (B) = [0.58204] A- Compute v from the equations, P' cos a APPENDIX XI 359 23. Then sin (S+v) 9 -T^ > em 6=g . A, 9 to be + or but less than 90. 24. tan M sin I = cos M cos &, tan h = -- : -. ami If *, ', be both less than s + appearance, A' = <(s + 6") o, (Annular) ( ff (s + 6"). 6" is added as a mean augmentation of s. 25. When n < P' A ' both limits will have place. When n > P' A ' only one limit will have place, viz. : 26. First and last points or places of entrance and departure : n A' /cP'\ cosw = =r, , r=| Isinw; TepaTre j = time of midd " | ~ \ ' Places of entrance and departure determined as in Nos. 4 and 6, for the begin- ning and ending of a phase, using a = ( i) w and 6 = ( i) + w. For the appearance of external contact these determinations are included ic No. 18, and therefore need not be repeated for these limits. 27. Places for any times within the limits : Prepare the following constants, using 6 at (5, A' sin i u = A cos , D = o T w, a = cos D' ' n n A' , E = -, cos w = =-,: as above ; c (n A') P 360 SPHERICAL ASTRONOMY. 28. Let t be the time from the middle of the general eclipse, Jf=(-OTa>'; 29. tan (A - a') = sin 9 cos (f + D') check . . . tan e = tan Z cos Jf, tan Jf, tan / = tan (e + D') cos (h a sin sin Z <:os M cos (6 + J) 1 ) cos (A a') cos I' < 90, and same sign as cos M ; and A a to be in the same semicircle with M. 30. For a more accurate determination at any time, Find P' = p (P *) for the latitude before found. Also, with Z find the augmented semi-diameter s'=s + augmentation, from the table annexed. Z Augmentation of the D's Semi-diameter. Argument : True Zenith Distance Z. For P = 54' Var. for 10' in P. Z ForP = 54' Var. for 10' in P. Z ForP = 54' Var. for 10' in P. o // ii n (/ o /, o i4-o 5. 7 3o I2 I 4-9 60 6-9 2.9 I i4-o 5-7 3i 12-0 4-8 61 6.7 2.8 2 i4-o 5. 7 32 11-9 4-8 62 6.5 2.7 3 i4-o 5-7 33 II- 7 4-7 63 6.2 2-6 4 i4o 5. 7 34 ii. 6 4.7 64 6.0 2.5 5 i3. 9 5. 7 35 n. 5 4.7 65 5.8 2.4 6 i3. 9 5-7 36 ii.3 4-6 66 5-6 2.3 7 i3.b 5. 7 37 II. 2 4-6 67 5.4 2-2 8 i3.8 5. 7 38 II -0 4-5 68 5-2 2-1 9 i3.8 5-7 3 9 10-8 4-4 69 4-9 2-O 10 i3.8 5-6 4o 10.7 4.4 70 4.7 9 ii i3. 7 5-6 4i io-5 4-3 71 4-5 8 12 i3. 7 5-6 42 10-3 4-3 . 72 4-2 7 i3 i3.6 5-6 43 10 -2 4-2 73 4-o 6 i4 i3.6 5-5 44 IO-O 4-i 74 3-8 5 i5 i3.5 5-5 45 9.8 4*i 7^ 3-5 4 16 i3.4 5-5 46 9.7 4-o 76 3-3 3 i? i3-4 5-4 47 9.5 3. 9 77 3-1 ' 2 18 i3.3 5-4 48 9'3 3.0 78 a-8 I -I 1 9 13.2 5-4 49 9.2 3-8 79 2-6 I I 20 i3.i 5-4 5o 9-0 3. 7 80 2-4 I 'O 21 i3.o 5-4 5i 8-8 3-6 81 2I 0-9 22 12.9 5-3 52 8-6 3-5 82 1-9 0.8 23 12.8 5-3 53 8-4 3-4 83 1-7 0.7 24 12.7 5.3 54 8-2 3.3 84 1,4 0-6 25 12.6 5-2 55 8-0 3-2 85 1-2 0-5 26 12.5 5-i 56 7 .'8 3.2 86 t'O o-4 27 12.4 5-i 57 7-5 3.i 87 0-7 o.3 28 12.3 5-0 58 7 .3 3-1 88 0-5 0-2 29 12.2 4.9 5 9 7-1 3.o 89 o-3 0-1 3o 12. 1 4.9 60 6-9 2-9 90 0-0 oo APPENDIX XI. 36J Then, f Partial J r s 1 + a, For j Total V phase, A' = J ' *, f Annular ) t a a? 81. For the time of d form the constants, (A) = [0.58204] ai cos D, (B) = [0.58204] Find the values of D, S, a, for the given time. (D) = D + (a corr. from table, page 342). () = (/>) -4, (y) = cos(J9), www r _. P' COS 5 82. (Z from the first computation), sin = /4/ cos Z tan . - , tan t = - 2 A cos v cos 2 ^ u = A ' cos t', D' = S T M, v = A' sin t', o' = , cos D a') corr. Remaining computation the same as in No. 29. VI. CENTRAL LINE. 38. The computation of the limiting times and places is comprehended undei the head, " Beginning and Ending of a Phase on the Earth." 34. Places for 3 any times within the limits: t = the time from the middle. t n tan w = -, A = - k c cos * w > 90 when n is negative. 85 = i) T; 86 (S at 6). sin Z = -^, tan = tan Z cos 8, tan h = T^X^: tan 8, tan / = tan (8 -f- b) cos A, cos (6 ~r i) 362 SPHERICAL ASTRONOMY. sin 6 _ sin Z cos S ' cos (0 + i) cos h cos I ' 8, same sign as cos S, and less than 90: A, same semicircle with S. 87. For a more accurate determination at any time, find P' t 8, A, as in No. 14, and proceed again with these as in No. 36. 88. Place where the eclipse will be central at noon : (**<$) Apparent Greenwich time of true (5 = longitude W. Z < 90 and same sign as diff. dec. 39. For a more accurate determination, find the horizontal parallax for the lati- tude, and with it repeat the operation. [All latitudes in the preceding formulae are to be recognized as geocentric, and will therefore need reducing by the table at page 336.] Examples. For an elucidation of the practical application of the preceding formulae, we shall take the solar eclipse of May 15, 1836. At the time of new moon, viz. 2*" 7 m -o. the moon's latitude is a5' 43", which being less than i a3' 17'', the eclipse is certain. (See the limits at page 333.) The elements of this eclipse, as related to the equator, are d. h. m. s. Greenwich mean time of <3 in R. A. . . . May i5 2 21 22-9 D 's declination ......... N. 19 25 9-8 's declination ......... N. 18 67 58-8 D 's hourly motion in R A ..... 3o 8 3 's hourly motion in R. A ...... 2 28 2 D 's hourly motion in declination .... . N. 9 68.7 's hourly motion in declination . . * N. 35 i D 's equatorial horizontal parallax ... 54 23 9 's equatorial horizontal parallax ... 8-5 I> 's true semi-diameter ....... , 1 4 49-5 's true semi-diameter ....... 1 5 49-9 from which we prepare the following values : Q D'sdec. . . +J925io D 's H. M. in R. A. . 3o 8 's dec. . . + 18 57 59 's H. M. in R. A. . 2 28 Diff. dec. . . + 27 ii o! . . . . 27 4o >'sH. 'eH. M. in dec. . M. in dec. * 9 5 9 35 D's 's Rel. eq. eq. eq. P hor. hor. hor. par. par. par. . 54 . 54" . 54 24 9 [5 10 log. const. log. 3-51255 9'999 2 9 3.5u84 Di .' f 9 24 APPENDIX XI. 363 I. BEGINNING AND ENDING ON THE EAKTH. A + 9' 24" .... 2.75128 (i) ai 27 4<> .... 322OII 9.53117 D -f 19 25'. 2 -cos . . 9-97456 j tan . . 9-55661 (2) <+I949 1 cos . . 9.97349 (3) diff. dec. -f- 27' 11" .... 3-21245 n + 25 34 .... 3.18594 sin t . 9-53oio (2) -f (3) const. . 3-5563o 6-27234 (4) e . . 3.52io6 (4) (i) t . . -f. 19 56 . & tan t . 3-07767 d. h. 6 . i5 2 21 a3 1 5 a i 27 middle of general eclipse P' . . . 54' i o''= A for central phase s -f P' A', these limits will extend throughout the whole duration 01 the eclipse ; and we may therefore calculate the position of a place for any time between the Greenwich times i4 d 23 h 6 ra 3o s , and i5 d 4 h 56 m 24 s . As an ex- ample, take the time i5 d o h 3o ra . 8 m Sm ,s'+m Assumed time . Time of middle . d. h. m. a. i5 o 3o i5 2 I 27 1 3i 27 3.73900 ' . 58 5i c . j tan . . ( cos . . n . . 3-52iot . 0-2182 . 9.7140 . 3.1859-. . 49 24 . 24 42 j log A . } comp. . 3.47191 . 6.62809 . 12 56 . 2-88986 . 17 43 . 3-02653 Comp. log P' . 6-48816 ' 17 0.9 sin -J m . 2)18-93264 . 9-46632 cos (S m) 9-58648 COS i . sin I . I . . Reduction Latitude . -- 9-97576 sin & . tan h . h . H . Longitude . + 9.51191 9.56224 . 0.86659 S. 21 24' 8 . 82 i5' 8 29 S. 21 32 i . W 90 44 PLACE FOLLOWING. h. m. a tan (S m) + o-3785o Greenwich time o 3o a Equation + 3 56 ( time o 33 56 ZTin ( space 8 29 APPENDIX XI. 365 PLACE ADVANCING. uos (S + m) . . + 985225 tan (S + m) . . 9.99444 cos 3 . . . . + 9-97576 sin i . . . . + 9.51191 tan h . . . . + 0.48253 h . . . . 108 i3 Reduction n H 8 20 Latitude . N 42 29 Longitude . . . "Vf. 116 42 By taking S = ( ) + w instead of ( <) w, similar computations will give the places following and advancing for the interval t = i h 3i m 27" after the time of middle, or for the Greenwich time i5 d 3 h 32 m 54 8 . Much time will be saved by taking the computations two and two in this manner. in. PLACE WHERE THE RISING AND SETTING LINES INTERSECT. o / 90 o a 18 58 71 , diff. dec. + P 1 . . co* i A' COB* sin P P IT . . . 9-99872 . . 3-51255 54' 5" . . 3.51127 sin S . . const. . . + 9.51191 . . 9-41796 + 4 36 . . + 2-44114 cosD . 3-22OII . . 9-97456 + 26 6 . . 3.19467 3o 42 = fi cos ' . ft sin t' . (tan . . + 3-26529 . . + 2.75128 + o485oQ '7 '' | C09 . . . . . + 9.98056 10 4o u . . + 3.28473 . 248 27' n" + 54 5 + 26 54 + 3 .20700 + 3.20842 9-99948 + 3-2o84a + 9-97349 sin* . . . + 953oio + 3-18191 3-26458 3-5563o f 6-29482 9.91733 \ OS f + 3*oiooo 366 SPHERICAL ASTRONOMY. A f sin . 3-OI4Q4 . 3- 284 7 3 <-.. T 2-95424 ^PP- time true d ii 42 56 at the place 8 4' . % . 2.68445 2 2i 23 cos ... '' 9- 99 568 Equation .... 3 56 tan J . . 9536i5 App. time true d . 225 19 at Greenwich tan I . ; j 0.45953 i time "5 ; Long, in ] I . . N. 70 5i ( 8 P ace Reduction . 7 Latitude. . N. 70 58 Thus we find the required place to be in longitude E. 1 39 a4' and latitude N. 70 58', where simple contact will have place at sunset and again at sunrise ; also the middle of the eclipse would be seen at midnight if it were not intercepted by th% opacity of the earth. The duration of the eclipse will correspond with the duration of the night, and therefore no portion of it will be visible. IV. PLACES WHERE THE MIDDLE OF THE ECLIPSE HAS THE SUN IN THE HORIZON. In the present case n is > P' (s + a) and < * + , We must therefore pro- ceed as in No. 19. 1. For the extreme points, 3.52io6 3.5n84 - . . 3. 84696 (i) + 25 34 ........ 3.i85 9 4 55 ....... 2-48430 . 8.97246 . . 9.99808 (2) * 56 3 8 9 . . . 3-845o4 (i) + (a) 2 i 27 time of middle a ir5i2 o448 time of beginning 6 + 5 34 3 58 6 time of ending APPENDIX XI. 367 PLACE OF BEGINNING, OR FIRST EXTREME PLACE. h. m. s. cos a . . cos S . sin I . / . . Reduction Latitude cos b . cos 6 . . sin I . . I . . Reduction Latitude 9.62918 tan a sin i . tan 7* h . H . + 0.32738 + 9.51191 Greenwich time Equation j time ( space PLACE. Greenwich time Equation . {time space o 4 48 3 56 9-60494 o8i547 ' 81 18 o 8 44 2 II' ' S. 23 45 8 + 2 II h. m. s. 3 58 6 3 56 S. 23 53 PLACE + 9.39664 Longitude W. 83 29 OF ENDING, OR LAST EXTREME tan 6- . +0.58943 sin i . . + 9.51191 tan h . 1.07762 + 9.37240 N. i3 38 5 4 2 2 60 3i' h H . Longitude ' + 94 47 60 3i N. i3 43 E. 34 16 2. For the extreme times, eP f the value of r x taken out from the preceding logarithm of > is i h $7 10* h. m. s. 2 i 27 time of middle i 57 10 . . TJ o 417 first appearance 3 58 37 last appearance PLACE OF FIRST APPEARANCE. sm i . coat . . sin I . . I . . Reduction Latitude + 953oro + 9.97576 cot i . sin S . tan h . . h . . II . Longitude + 0.44339 + 9.51191 9-5o586 0-93148 S. 1 8 42 7 83 19 + 23 S. 18 49 W.85 22 PLACE OF LAST APPEARANCE. Latitude N. i84 9 ' 83 19 1 80 o h . H . + 96 4i + 60 38 fa. m. 8. Greenwich time o 4 17 Equation . 3 56 ( time o 8 i3 Hint ( space 2 3' Longitude K 36 3 h. m. s. Greenwich^ time 3 58 37 Equation ~. . 3 56 {time 4 2 33 -r-rTcf space oo 38 368 SPHERICAL ASTRONOMY. For the computation of places in this line, we have therefore the whole rang* between the Greenwich mean times o h 4 m i? 8 and 3 h 58 m 37*. As an example, take the time i h 3o m . h. m. B. Time of middle 2 i 27 i 3o t . . o 3i 27 3.2 7 5 77 (0 c -! . i o 4o n 3-84696 a 1 5 34 . sin 9.42881 a . b . . . 35 23 . . 4 i5 h. m. s. cos a . + 9.91132 tan a . . _ 9 .85r4o Greenwich time i 3o o cos 8 . . + 9-97576 sin S . . + 9.51191 Equation 3 56 sin I . + 9-88708 tan h . . + 0-33949 {time i 33 56 I N 5o 27 h o / n4 35 space 23 29' Reduction ii H . + 23 29 Latitude N. 5o 38 Longituc le W. i38 4 By similarly using the angle b, we shall find the position for the interval 3i m 27' after the time of middle, or for the time 2 h 32 m 54 s ; thus, cos b . cos S . sin / . Reduction Latitude + + + N. 9.99880 9-97576 tan 6 . sin S tan h . h . H . 8 . +9 +9 87106 .51191 .35915 Greenwich time Equatior itime space 2 3 a 3 54 56 9 97456 2 36 5o , 70 35 7 o / 67 7 39 i3 3 9 i3 N. 70 42 T ftn _; fn j- j W. 206 20 The places may be computed by two together in this way ; and it will perhaps be a little more convenient to assume a value of t in the first instance. We may take any value which does not exceed r l or i h 57 m io 8 . In the present example we should take t= 3i m 27 s , and begin as under: ) . a . b . ' . 19 49 i5 34 log* sin co ' cP' n 3.27677 3-84696 . 35 23 . 4 i5 9.42881 and then proceed for the places as above. h. m. s. Time of middle .2127 t . . . 3i 27 Time before middle i 3o o Time after middle 2 32 54 APPENDIX XI. 369 V. NORTHERN AND SOUTHERN LIMITS. 1. FOR THE PARTIAL PHASE, we have only southern line of simple contact. Constants E, cos w, D', a'. 1 4' 56" *+6" i5 5o A' . . n . n A' . 3o 46 + 25 34 ... + 3.18594 2. 4941 5 3-5n84 5 12 ... c . E . . A' . COS t log w . u . .5 . D' 0-69179 3.52106 P' . . 7. 17073 cos w 3.26623 . . . . + 9.97349 sin i 8.98231 3-26623 + 953oio + 3.23972 28' 5 7 " cos.Z>' . o + 1 8 57 59 log a' + 2-79633 + 9.97448 2-82185 + 19 26 56 a' . ii' 4" The extreme places will be the same as those which have the middle of the eclipse with the sun in the horizon, page 366 ; and we mav compute for any time between the corresponding times of beginning and ending, viz. : o h 4 m 48 s and 3 h 58 m 6 3 ; or we may take any value of t less than i h 56"' 39*. For an example, take t = o' 1 58 m 33 s . 3 ' 5 4568 7-17073 Time of middle t Before middle After middle h. m . 3. 2 i 27 58 33 1 2 54 3 o o 19 49 100 53 E M 120 42 -f 81 4 + 3o35'-3 tan COS U) COS W sin Z tan Z Remaining calculation for the time 3*> o m o 8 . e+ 5 i4-7 D' + 19 26-9 jy + T4 4i-6 a ' + 32 37-2 ' 1 1 I + 9-77167 + 9-19113 + 8-96280 + 8-96098 + 9-95835 32 26-1 tan Z . . cos M . , tan e . . sin e . . cos . . . tan M . . + Q.8o357 tan . . . + 9-80620 cos . . . + 9-92544 tan (e + D r ) + 9-66258 tan / sin Z . cos M 0-71641 9-27571 8-98231 -f 9-70660 + 9-77167 + 9-70660 + 9. 19113 Redaction Latitude + 9.58802 o , N. 21 10-2 7-6 N. 21 18- + 8-89773 Comp. cos (h a') + 0.07456 Comp. cos / . . + o.o3o35 check + 9-00264 h. m & Greenwich time 3 o o . 3 56 Equation time . 3 56 24 space + 45 59' h . . . +32 26 Longitude . W. i3 33 370 SPHERICAL ASTRONOMY. The calculation for the time i h a m 54* is to be performed in this manner, with the same values of tan Z, sin Z, only taking the value of M = 120 42'. A MORE ACCURATE CALCULATION FOR THE TlME 3 h O m O 8 . Constants (A\ (). i . . . . + 3-220II cosD . . const. . . 9-97456 A . . . + 2.75128 o582o4 3.77671 3-33332 (^1) . . . + i3 9 '4o" (B) . . These constants may serve for the computations at example the following is the process employed : O 1 II t II D . 4- 19 3r 34 \(j)\ a + l l 49'O a corr. . i i t ; i . f- 18 58 21 a . . + 3.02898 (x) . . t- o 33 i4 cos(D) +9-97428 . + o 35' 54" all times. For the present P * .. . 3-5i255 p . . 9.99982 log (*) - ftin S (x) sin 3 (A). . P' cos 5 X cos v 2 - 2 X COS V 9 HUg. . ' f . A'. f 3-29973 L- 95i2o4 . log (' . + 18 57 59 + 1.26249 . + 9.97579 + 1-28670 . + 18 57 8 a' . . + o o' 19" The semi-duration of the northern limit on the earth is therefore i h 42 m i4', and we may calculate for any value of t not exceeding this. A calculation of the extreme places on the earth is to be performed the same as for the beginning and ending of a phase on the earth, and will bo unnecessary here. As an example, for a time within the limits, we shall take t = i h io rn o 8 . h. Time of middle . 2 t . . i m. s. I 27 (- IO O *' . . 19 49 E . 5o 43 tan u>' . 3-62325 + 6-464o3 + 0-08728 Before middle . o 5i 27 . . M . . A fter middle . 3 1 1 27 . . M . . + Z . . + Remaining cahulation for the time 3 h i i m 27". tan Z . . + o-o8423 cos M . . + 9-93352 70 32 cos w' . 3o 54 cos w . , j sin Z . 5o 3i-3 ] 7 ( tan Z . sinZ. . . . cosJ/ . . . + 9-80147 + 9-68901 + 9-88754 + 0-08423 + 9-88754 + 9-93352 6 + 46 IO-2 tan & + 0-01775 + 9-82106 J)' + 18 5 7 -i sin e . + 9-85817 comp. cos (h a) + 0- l5622 + D' + 65 7-3 . cos . + 9-62397 com p. cos I . + 0-25693 + 0-23420 . check + 0-23421 tan M . . + 9.77706 o ' ( tan . . + 0-01126 A - j ' I jf 5 44.6 J i. a . + o.3 ( cos . . + 9-84378 Greenwich time 3 n 27 A . + 45 44o tan (8 + L ;') + 00,3,374 tan / . + 0-17752 {time . 3 i5 23 / . K56 2 3'. 8 space + 48 5i' Reduction 10 4 h ... + 45 45 Latitude . N. 56 34 Longitude W. 3 6 The calculation for o h 5i m 27" is to be performed in the same manner, with = 70 3a'. APPENDIX XI. 373 A MORE ACCURATE CALCULATION FOR THE TlME 3 h II 1 O 2 7 . D . . a corr. . J . . (*) - + '933 a?l (I)) . . + *8 58 28 a . . + o 35 o cos (Z>) + a3 6 + 3-14176 + 9-97419 P T . f) . . 3-5ia5S 9.99901 log (x) . sic 3 + 3-32222 log(y) + 95i2o8 + 3.ii595 + 9.51208 P' . . cos S 3.5n5ft 9-97574 + 2.8343o + 2.62803 FcosS 3-48730 (x) sin 5 U). . + n' 23" (y) sin 3 + i 3 9 4o (5) . + o i 5" + o 35 54 cos Z . 2 X COS v 9-8o33i O-6374** j + i 5i 3 j + o 28 49 sin 2 . 9-16591 (log . + 3-82367 [log. + 3-23779 sin ^ 9-58296 p cos i 3.48730 3.48730 (i 22 3o'4 ^ COS v + O.33637 * sin " + 9-75049 2? . . 45 o -8 2 o3oio3 X cos v - + o.3363 7 COS 2 . 9.84989 j A COS V + o 63740 tan v 4- o AiAi a 4- o*4{ 4i 2 ' . . + 20 9'. 4 . / tan*' . ) cos/ . . + 9-56473 + 9.97255 4 . . aug. . < // . i4 5o . . 9 { sin 4' . . + 9.53-73.8 ' . . i4 5 9 i5 5o A . o 5i .... I 7075? I *7O75'7 cos *' + 9.97255 sin t ' . . + 9.53728 + 1-68012 + 1-24485 w , S ... + o o'48" + 18 58 28 cos D 1 . + 9-97577 + i 26908 D 1 ... + 18 57 4o a' . . a . . . -f O O' 19" . + o a3 6 D . . . + 19 33 27 (a_a'. . + 22 47 ( o') corr. i (log . . +3- 1 35 7 7 3 */ . oo e_> T !tan + 9'7779 3 M . + dc 07 i . cos . + 9.93328 P 1 + 3.398 7 5 . +3-5u56 J . . . + 5o 27'. 9 lain . ( COS . + 9.88719 . + 9-8o383 374: SPHERICAL ASTRONOMY. . +46 . + 18 7+65 ' + 45 4i + o t 5-8 57-7 3-5 9 - 4 cos M . . tan e . . sin ^ ff . cos . tan M . {tan . . . cos . tan (0 + D'} tan / . . Reduction . Latitude . + o-o8336 + 9-93328 smZ . -*. cos M . -1- 9-88719 + 9-93328 + 0-01664 + 9-85764 + 9-62499 + 0-23265 + 9'7779 3 + o.oro58 + 9-84412 + 0-33249 + 9-82047 comp. cos (h a) + o-i 5588 comp. cos 1 . + o-2563o . check . . . Greenwich time Equation . {time . space h . . Longitude + o 23265 h. m. a. 3 ii 27 3 56 + 45 4a 3 + o i 766 i + 3 i5 23 K56 20'- 5 10 -4 + 48 5o'.8 + 45 42 -3 N.56 3o -9 W. 3 8-5 VI. CENTRAL LINE. "We have, at page 363, found the semi-duration of the central appearance on the earth to be i h 43 m 17*, which is therefore the greatest value of t for this phase. As an example for a time within the limits, take the same value of t as in the two preceding examples. , 3.623*5 , 3.52io6 Time of middle t Before middle \fter middle b. m. s. 2 I 2 7 o 5i 27 3 ii 27 19 49 + 5i 4i 71 3o + 3i 62 Remaining computation for the time 3 h u ra 27 s . 6 + 44 56-o i + 18 58-0 + 6 + 63 54-o + 44 56-6 tanZ . cos . tan e . sin 8 . cos . tan 8 . j tan h . ^ cos h . tan (e + i] tan / . I . Reduction Latitude + 0-06994 . + 9-92905 . + 9.99899 . + 9.84898 . + 9-64339 + o 20559 . + 9.79354 . + 9-99913 . + 9-84991 ). + 0.30990 . +0-15981 . K55 i8'- 7 10 .6 . N. 55 20. - t , e tan COS u, ii . A . P" . 0-10219 9-799.46 sin Z. . cos 8 . comp. cos h comp. cos I check . 3.3 9 348 . 3.5ii84 . 9-88164 . 0-06994 + 9-88164 + 9-92905 + 9-81069 + o- i 5009 + 0-24480 o2o558 h. m. s. Greenwich time 3 1 1 27 Equation . 3 56 ( time space h . . . Longitude + 48 5r + 44 5 7 W.T 54 APPENDIX XL 375 A MORE ACCURATE CALCULATION. D . a corr <*) - 4- IO 33 27 ) / Tvt , . + 23 6 . + 3.14176 *y A3 *j f / m 1 4 cos 2) . . . . + 18 58 28 / V + 3.ii595 + (x) . + 3.32222 3-51255 8 . +3i5a'- 7 | ; tan S . . . ! cos S . . . + 9.79373 + 9- 92899 9.99903 A .... + 3.39323 3.5n58 . . . 3-5n58 Z tanZ . . + 49 35' -6- . + 0-06994 [ sin Z . . . [ tan Z . . . smZ . . . + 9.88165 + 0-06994 + 9.88165 o cos S . + 9.92899 cos S . . + 9'9 28 99 + 44 55-8 tan . + 9-99 8 9 3 + 9. 81064 + 18 58-5 sin . . + 9 .848 9 5 com p. cos h + O I5O20 + 63 54-3 . cos . . . + 9-6433i com p. cos I + 0.24480 + o-2o564 . check . . . + o-2o564 tan S . . + 9-79373 o 4- 44 5 7 .5 I"" 1 * ' - + 9-99937 ( cosA . . + 9.84980 Greenwich time 3 ii 27 tan (0 + 3 ) + o-3iooo Equation . 3 56 tan/ . + 0-15980 ( time . TT ] + 3 i5 23 Z . . . N. 55i8'.7 ( space . + 485o'.8 Reduction 10 6 h . . . . + 44 5 7 .5 Latitude . N. 55 29 .3 Longitude . . W. 3 53.^ CENTRAL ECLIPSE AT NOON. Diff. dec. P . smZ . Z . t . . / . . Reduction Latitude . 3.21245 . 3.5n84 . 9.70061 Time of 6 Equation . . | time . Long, in { ( space h. m. 8 . 2 ai 23 + 3 56 2 a5 19) (. w + 3o 8' + 18 58 36 20' ( N. 49 6 ii N. 4 9 17 By assuming a series of times, and so computing, in conformity with the preced ing examples, a series of points on each of the several limits will be determined; and these points being laid down in a geographical map, with respect to latitude and longitude, it will be easy to trace the lines through them. In this manner has the following map been executed, the assumed law of projection being that the parallels of latitude are concentric and equidistant circles. This projection will be found ver^ suitable when an eclipse, as in the present instance, extends com- pletely round one of the poles of the earth. In other cases, any hypothesis what- ever may be assumed, with respect to the law of projection, provided the geo- graphical sketching and eclipse-lines be both laid down on the same principle, (See Fig. 11.) PRINCIPAL LINKS FOR THE SOLAR ECLIPSE OF MAY 14-15, 1836 lass s g ' *' s s s APPENDIX XL 377 PHENOMENA FOR A PARTICULAR PLACE. I. ECLIPSES OF THE SUN. The chief objects of determination for any particular place are 1. For a partial eclipse, its magnitude, and the times of beginning, greatest phase, and ending. 2. For a total eclipse, the times of external and internal contact of limbs, or the times of partial and total beginning and ending. 3. For an annular eclipse, the times of exterior and interior contact of limbs, or the times of partial and annular beginning and ending. Also, to secure certainty in the observation, it is necessary to determine, in each case, the particular points on the limb of the sun, as related either to the vertical or a circle of declination, where these contacts take place ; and hence the general configuration of the ellipse. We first proceed to find expressions for calculating, at any time, the apparent relative position of the two bodies, and the augmentation of the semi-diameter of the moon. The parallax in altitude depends on the Eq. (8) or (9), page 336. It will here be necessary to investigate the effects which this parallax will produce in the right ascension and declination of the moon. These might be accurately determined by the theory of the small variations of spherical triangles, but not quite so simply as in the following manner: Assume, as before, I, the geocentric latitude of the place ; R. A., the true right ascension of the moon; D, the true declination of the moon, + north, south; h, the true hour angle of the moon, + west, east ; r, the distance of the centres of the earth and moon. Then if, from the earth's centre, we take a, on the intersection of the planes of the meridian and equator, -t towards upper meridian ; y, in the plane of the equator, + west, east ; z, parallel to the earth's axis, + north, south ; we shall have, for the position of the moon, x = r cos D cos h, y r cos D sin h t z = r sin D ; and, for the position of the observer, (a;) = f> cos I, (y) = 0, (z) = p sin I. Thus the position of the moon, in relation to the observer as an origin, will be ' = x (a;) = r cos D cos h p cos I ; y' = y (y) = r cos D sin h ; z' = z (z) = r sin D p sin / ; and hence, D\ h! denoting the apparent declination and hour angle, and f f Ihtf distance of the moon from the observer, we shall have x = r' cos D' cos h' = r cos D cos h p cos / ; y' = r' cos D' sin h' = r cos D sin h ; ' s= r' ain D' = r sin D p sin I. tt) 378 SPHERICAL ASTRONOMY Therefore, as cot A' = - , tan J}' = - sin h', - = sin P, we find y y r p sin P cos I cot A' = cot A =r r- cos D sin A / p sin P sin /\ sin h 1 tan Z>' = I 1 " : =. I -. r tan J) \ sin D / sin A or / p sin P \ cot A cot A' = I 77: : I cos I \co8 D sin A/ tan D _ tanZ)' __ / p sin P V ^ sin A sin A' \cos .Z) sin A/ which present a direct method of calculating the apparent position of the moon, at any time, from that of the true. The former of these equations is evidently subservient to the other, and must necessarily be computed first. As the calcula- tion of these expressions will, in general, require seven places of figures, it will be more convenient to determine the simple effects of the parallax, or the small dif- ferences A.R. A.R.' t D D', for which other expressions may be derived from them. Let A.R. A.R.' = A' A = A A, and D D' = A D ; then by multi plying the equation . . . p sin P cos / COt A COt A = ; r- cos D sin A by sin A sin A', the left-hand member will become sin (A' A) or sin A A. p sin P cos / . . . sin A A = ^r sin A . cos D Again we have tan D tan D 1 p sin P sin / sin A sin A' cos D sin A ' But tan D tan D' tan D tan D' sin A sin A' sin A \sin A siu A' _ sin (D D') _ sin h' sin A ^ sin A cos J9 cos D' sin A sin A' sin A 7> 2 sin A A cos (A + i A A) 4 _ . ^ H * ; r :-; tan if . sin A cos D cos D' sin A sin A A , , p sin P sin Z Equate this with - -- 77-7 r, and we find cos D sm A sin A -D _ p sin P sin / 2 sin A A cos (A + | A A) sin cos Z> cos D' cos Z> sin A sin A A p sin P cos I sin A' But 2sm i /. A = APPENDIX XI. 379 Substitute this value and multiply by cos D cos D' and we deduce sin A D = p sin P I sin / cos D' cos / sin D' I. L cos i A A J We shall therefore have, for the parallax of the hour angle, and that of the decli- nation, (p cos Z) sin P . 'J sin A h = - sin h' cos D I" (2) sin A D = sin P I (p sin I) cos D' (p cos f) sin D' - ^ I These are still however not adapted for direct calculation, since they involve the apparent quantities A', D', which it is our object to determine. The only use that can be made of them is, first to use the true quantities, in order to get the parallaxes and apparent values approximately, and then to repeat the operation. To avoid this difficulty, substitute in the former A + A A instead of A', and in the latter put J) AD instead of J)', and we get, by expansion, p cos / sin P . . sin A A = (sm A cos A A -f- cos A sin A A) ; sin A D =p sin P cos AD I sin / cos D cos I sin D ' I L cos i A A J 4- p sin P sin A D \&a\ I sin D -f cos I cos D - 1 t-"L__J I L cos J, A A J Divide these by cos A A, cos A D, respectively, and solve for tan A A and tan A 1), and we find /p cos / sin P\ , I - - 1 sin A V cos D / tan A A = . (8) /p cos I sin P\ 1-1- j, 1 cos A V cos D / tan A D P sin P Fein / cos D - cos / sin D cos ( h + JL L cos \ A 1 p sin P I sin I sin D -f- cos / cos D cos i A A ^ T tanD cos (A + | A A)~l ' " (p sm / sin P) cos D I 1 . J L- I L tan / cos -J A A J / 7 nx r^r, 1 COS (A 4- i A A)"l 1 - (pain /sm P) sin D\ 1 + . S L tan / tan D cos i A A J These expressions are all of them perfectly rigorous, and better suited to calcu- lation than they would appear at first sight. The process of the calculation, in which. five places of figures will be sufficient, is more detailed in the following equations: (p cos 1) sin P n sin A n = L - ; tan A A = . . (5) cos D 1 n cos A v ' 380 SPHERICAL ASTRONOMY. c = (p sin /) sin P ; n x = k tan D ; eou tan AD = n a = c cos D (1 k tan D 1 c sin D (1 -f n a ) The expression (4) for tan A -D may, however,. be neatly resolved by means spherical triangle as follows: Assume Fig. 9. of (A) being very nearly equal to h -{- A #. And let N be the north pole, Z the central zenith, and AT the moon; then NM = 90 D, NZ = 90 J, and the / N = h. "Without changing these values of NM, NZ, let us suppose the hour angle JVto become increased to the value of (A); and with the triangle so constituted suppose the altitude of the moon to be , so that ZM= 90 ; then the spherical relations will give sin Z M cos M = cos NZ sin N M sin NZ cos N M cos N, = cos NZ cos NM + sin NZ sin NM cos N, cos cos M = sin J cos D cos ^ sin D cos i T^ i T-K = sin I cos JD cos / sin .Z) 4- $ A cos A sin e = sin / sin D -\- cos / cos D cos (^) . , . r, , r, cos (A + -i A A) = sin / sin D + cos I cos .D -- ^ - :. cos -J A h Comparing these with the former expression of (4), we have therefore tan A D (p sin JP) cos e . cos M 1 (p sin P) sin t Before this can be used the angles M and e must be determined. Draw ZD perpendicular to MN, and by spherics, tan ND = tsn\NZcos N . . . . ^^ , sin M D tan M = tan ZD = sin N J) tan N; Also by (c) sn . tan M = - - tan N sin M D ta.nMZ = , or cot MZ = cot M D cos M cos M tan M cos N sin M sin MD ~~ tan cof' sin 7 cos N sin NZ co jTsin MZ (d) APPENDIX XI. 381 Let now ND = 0, and M D = MN = 90 - (0 -f D) ; and the equations (a), (6), (c), (d), (4 (/), will give the following: __ cos (A -f \ A A) ( a \ tan cos - A A ~~~ cot 1 cos (A) . . . r . (b) 8in * tan (A) (e\ tan e cos (0 + D) tan * ' tan (0 -f- D) cos 3f . . . . . (d) sin cos (h) cos I (e\ cos (0 -f- D) tan A D cos Jlf cos (p sin P) cos e ( f} 1 (p sin P) sin e V/ ) in which the equation (e) is used as a check on the preceding computations. This check affords a good security to the accuracy of the work, and gives to these equa- tions a decided preference over those of (6), although a trifle more perhaps in point of calculation. They have also another advantage, inasmuch as M may be consid- ered as the parallactic angle, and c the altitude of the moon ; the former of these is useful in determining the position of the line joining the centres of the two bod- ies in relation to the vertical, and the other is useful in finding the augmentation of the moon's semi-diameter, which we shall now consider. If s' denote the moon's apparent semi-diameter, and s her true semi-diameter as seen from the centre of the earth, the actual semi-diameter of the moon will be represented by both r sin s, and r sin s' ; also, if a perpendicular be drawn from the centre of the moon upon the radius p produced, this perpendicular will be rep- resented by both r sin Z, and r sin Z '. We must therefore have = . sin s sin Z Let M be the true position of the moon, in the preceding figure, and sin ZM sin /. NZM sin NM sin N will be sin Z sin/ NZM=cos D sin A; for the apparent position of the moon the angle N Z M will remain the same, and sin 22 sin Z NZM=co3 D' sin A'. sin Z' .__ cos D' sin A' ' ' sin Z cos D ' sin A ' Also, by means of the equations (8) and (9), page 336, sin Z' p sin P sin Z' _ sin z cos z _ cos z sin Z f sin P sin Z p sin P sin Z ~" p sin P sin Z "~ 1 p sin P cos Z m sin s' sin Z' cos J)' sin A' cos z "sin x sin Z cos D ' sin A 1 p sin P cos Z All the preceding formulae are strict in theory. It now remains to consider what allowances may be made and what facilities given in their actual calculation. In the first place the value of cos A A may be safely assumed equal to unity, and may therefore be rejected in the equations (2), (4), (6), and (7), so that (A) = A + i A A ; it may be shown that this supposition cannot involve an error of more than 0".03 in the value of A D. SPHERICAL ASTRONOMY Also, as the arcs P, A A, A D, are small, we must have very nearly ^ = sin 1" = [4.68657], *!LA* = ta ^ J = tan 1" = [4.68657], where P, A h, A D, denote respectively the numbers of seconds they contain. These equations may be made more exact, for the limits between which the angles are always comprised, by adopting numbers differing a little from sin 1" and tan 1"; thus, by assuming ^ = [4.68655], '-^ = [4.68561]. Jr A A The first supposition will not in any case involve an error exceeding that of 0''.05 in the value of P, nor the second an error of more than 0".l in the value of A A, and these are much too small to merit attention ; the latter assumption ap- plies equally the same to A D. Thus we shall have (A) = h + i A h, sin P= [4.68555] P, A A = [5.31439] tan A A, A D = [5.31439] tan A D ; also, A h = A a, the parallax in right as cension. The equations (3) and (7) may therefore be commodiously arranged as follows : c = [4.68555] p ; A = c P ; m = A cos / ; . k = cos D f . . . (9) n = kcosh; A = [5.31439] ^^J By taking h less than 180, positively or negatively, A a will have the same ign as h. tan = cos (A) cot I ; G = cos (A) cos I tan M = , , y.. tan (A) ; tan c = tan (0 -f D) cos M \ ^ . (10) sin G 5 = cos J/cos e. check . . . , = - cos (0 + jD) A /? wi = ^1 sin t ; A 7) = [5.31489] 1 Wi The auxiliary arc may be taken out in the first quadrant, -f- or ; calling to 180 the first semicircle, and 180 to 360 or to 180 the second semicircle, the parallactic angle M must be taken out in the same semicircle with A ; and A D will have the same sign as cos M. It will appear by the preceding investigations that the values of A a, A D, so deduced, are the quantities to be subtracted from the true values of A.R., D, to get the apparent. As the number n is always very small, the values of comp. log. (1 w) to the fifth place of figures may be comprised in the following useful Table under the title of Correction of Log. Parallax, and conveniently taken out with the nearest third fig-ire of the argument. APPENDIX XI. 383 Correction of Log. Parallax. Argument: log. n. Log n Corr. Log n Corr. Log n Corr. Log n Corr. 1 Logr* Corr. 5-00 o 7'ioo 54 7.400 109 7-700 218 8-000 436 10 no 55 4io 112 .710 223 010 447 20 I i 20 5 7 420 u4 720 229 O20 457 .3o I i3o 58 43o 117 7 3o 2 34 o3o 468 .40 I *i4o 60 44o 1 20 . 7 4o 240 o4o 479 5o I i5x> 61 45o 123 7 5o 245 o5o 490 .60 2 .160 63 46o 125 760 25l 060 5oi .70 2 170 64 .470 128 .770 25 7 070 5i3 80 2 i8o 66 48o i3i .780 263 080 525 .90 3 190 68 490 1 34 .790 269 090 53 7 6-00 4 200 69 5oo i3 7 800 2 7 5 IOO 55o 10 6 210 7i 5io i4i 810 281. 110 563 20 7 22O 72 520 1 44 820 288 I 20 5 7 6 3o 9 .230 74 53o 1 48 83o 294 i3o 5oo 4o ii 24O 76 54o i5i 84o 3O2 i4o 6o4 .5o i4 25o 77 55o i55 85o 3o8 i5o 618 .60 17 .260 79 56o 1 58 .860 3i5 .160 632 70 22 .270 8k 5 7 o 162 .870 323 170 647 80 27 .280 83 58o 1 65 880 33i 180 663 .90 34 .290 85 5 9 o 169 890 338 -190 678 7.00 43 .3oo 87 600 i 7 3 900 346 200 6 9 4 7-000 43 3io 89 610 177 910 355 2IO 710 010 44 32O 9 1 620 181 920 363 22O 727 020 46 33o 9 3 63o 1 86 . 9 3c 3 7 i 230 744 o3o 47 34o 95 64o 191 .940 379 240 761 o4o 48 35o 98 65o i 9 5 95o 388 250 779 o5o 49 36o IOO 660 199 .960 398 8-260 79 & 060 5o 3 7 o IO2 670 204 .970 407 070 5 1 38o io4 680 20 9 980 4i7 080 52 3 9 o 107 690 213 7.990 427 090 53 7.400 109 7.700 218 8-000 436 7.100 54 This correction is additive when n is positive, and subtractive when n is negative. For the parallax in declination it will always be additive if the moon be above the horizon. For the augmentation of the moon's semi-diameter we may assume cos z = 1 and Z = 90 c, so that 1 p sin P sin 1 ni ' ni being the number which enters into the computation of A D. Hence , _ s __ [9.43537] P (H) 384: SPHERICAL ASTRONOMY. This and the last formulae for A a, A -Z>, entirely preclude the necessity of having recourse to a table of the sines and tangents of small arcs, and possess much uni- formity and simplicity in their application. To get the relative parallax of the moon with respect to the sun, we must use P IT, instead of P. If, therefore, P' denote the value of p (P TT), or the rela- tive horizontal parallax reduced to the latitude of the place, we must use sin P', instead of p sin P, in the preceding formulae. The determination of the apparent relative positions of the centres of the two bodies, as well as the augmentation of the semi-diameter of the moon, at any time, has now been reduced to a practical and expeditious set of formulae. A series of these apparent positions of the moon, with respect to that of the sun, will trace out her apparent relative orbit; and the contact of limbs will evidently take place when the apparent distance of the centres becomes equal to the sum or difference of the semi-diameter of the sun and the augmented semi-diameter of the moon. For a distance equal to the sum of these semi-diameters we shall have partial be- ginning or ending; for a distance equal to their difference we shall have Insular \ be g innin g or endin S> when '{<:[* Since the hour angle of the bodies is subject to the rapid variation of nearly 15 per hour, the effect produced by parallax will be of so irregular a nature as to give a decided curvature to the apparent relative orbit of the moon. This curva- ture will be more strongly characterized when the eclipse takes place at some distance from the meridian or near to the horizon ; and the apparent relative hourly motion of the moon, even during the short interval of the duration of the eclipse, will, through the same irregular influence, experience considerable varia- tion. These circumstances will, in some measure, vitiate any results deduced in the usual manner, by supposing the portion of the orbit described during the eclipse to be a straight line, and using the relative motion at the time of apparent conjunction as a uniform quantity. The method we are about to pursue is very simple, and consists in assuming any time within the eclipse, and computing for this time the relative positions and motion of the bodies, and thence finding, with- out any reference whatever, either to the time of the middle of the eclipse or to the time of conjunction, the times of beginning, greatest phase, and ending, and the relative positions of the bodies at these times. The nearer the assumed time is to the time of the greatest phase, the more accurately will the time of that phase be determined ; and, similarly, the nearer that time is to the time of begin- ning or ending, the more certainty will attach to the determination. To find the apparent relative motion of the moon, we must first determine the variation which takes place in the parallax. For this, take the equations (2), p, 879, viz.: . sin P' cos I . , * sin A a = sin A h = sin h, cos D sin A D = sin P' Fsin / cos D' - cos / sin D' (* + * ^ *)"| . L cos i A h J or, substituting small arcs instead of their sines, A D = P' [sin / cos D' - cos I sin V (k + i **>]. cos i A h J APPENDIX XI. 385 Since a portion of the apparent disk of the moon is projected on that of the sun, the apparent declination D' can differ very little from &. As the hourly variations of these small quantities are only required approximately, we may therefore use o instead of D 1 and neglect A A, so as to have r,, cos I . A a = P -- - sm h, cos D A D = P' (sin I cos S cos I sin <3 cos A) ; which vixpressions, though rough values of A a, A D, will give their hourly varia- tions pretty accurately. For these, observing that h is the only quantity which, by its rapid variation, has any sensible influence on these values, we have by differentiation, dt a) / dh . \ cos/ ' = I P 1 sin 1" 1 - - cos h, \ dt / cos D /r>,dh . ,,\ = IP' -j- sm l' 1 ) cos / sm t sin h. Bat by the equations (9), m = [4. 68555] P' cos/, = [4.68*55] P' -^- cos A. cos 2> Suetitute, therefore, P'^LL cos * = [5.31445] , cos D P' cos / = [5.31 445] m; ^A- } = [5.31445] (^ sin 1") m sin I .in . If we adopt 14 29' as a mean value of , we shall have sin 1"= [9.40274] and [5.31445] (^ sin 1") = [4.71719] or [4.7172]. Therefore, if ( = the time of describing In M s. with a < negative > sign. ( 2 ) {JSM} (positive) To have these times expressed in seconds, assume (w yj cos and then we shall derive / 1= =cein [ (8+ ], fc (f) = c cos b> sin [ (S 4- ], and hice ' beginning J ( c sin [ (5 + t) ] ) The time of < greatest phase > = assumed time -f- < c cos w sin [ (8 + t) ] V (18) ( ending ) ( c sin [ (8 + i) + ] ) It has been observed, that any one of these values will be the more to be de- pended on the more nearly it approximates to the assumed time. Thus, if the assumed time be within ten minutes or so of the end of the eclipse, the point M will approximate so closely to the point E, that no sensible error can arise by supposing the small portion ME of the orbit to be a straight line, and to be passed over by the moon with a uniform motion. This circumstance renders it advisable, in the first instance, to take the assumed time near to the time of the middle of the eclipse, so as to give a good result for the time of the greatest phase, and results for the times of beginning and ending, which may be nearly equally relied on. Such a computation will be sufficiently exact for the usual purposes of prediction. When the time of beginning or ending is wanted to great minute- ness to compare with observation, it will only be necessary to repeat the operation for ft time assumed as near as convenient to the first determination, which will mostly give within a fractional part of a second of the true theoretical result ; a degree of accuracy, however, seldom wished for, and quite unsupported by the present state of the lunar theory. To fix on a time near to the middle of the eclipse for the radical computation, one of the most simple expedients will be to determine roughly the time of the apparent conjunction. .TY i APPENDIX XI. "We shall now briefly consider the apparent positions of the moon, as related to '.he sun's centre. It is clear that S is the angle of position of the moon's centre from the north towards the east, at the time assumed ; also that the angle N~ S = u> -f- * is the similar angle of position from the north towards the west at the time of begin- ning; and that the angle N S E = u> t is the angle of position from the norih towards the east at the time of ending ; and that the angle N Sn = i is the same angle towards the west at the time of the greatest phase. Therefore, by estima- ting all these angles towards the east we shall have f beginning J f ( - ) - w J At ^greatest phase > / of J) 's centre from N. towards E. = j( i) V (19) ( ending ) ((- i) -f- w) In the computation of the parallax in declination, we find an angle M, which iu practice may be supposed to be the angle N 8 Z for the assumed time, the zenith Z being reckoned towards the east; consequently, at this time we shall have SJM for the angle of position of the moon's centre from the zenith towards the east. At any other time the parallactic angle J/for the latitude of Greenwich may be taken from the following table, arguments the corresponding apparent time and the sun's declination. This table, for any other place, may be computed by for- mulae, such as at page 381, viz. : tan 9 = cot I cos A, tan M = ; r- tan A, cos (9 + <*) A being the angle answering to the apparent time. Those who may be engaged in the computation of eclipses, for any particular places, will fiud considerable facility in the formation of similar tables. For an occultation of a star by the moon, the argument, instead of the apparent time, will be the star's hour angle, or the sidereal time minus the star's right as- cension. In this case the required positions will be those of the star with respect to the moon's centre, which will therefore be different from the angles of position for a solar eelipse, in which the moon's centre is referred to that of the sun. The angular positions of the contacts at immersion and emersion will consequently be determined in the same way as for an eclipse of the sun, and will be estimated in the opposite directions. Thus, for an occultation, And so must 180 C be applied to the other angles of position, as expressed for a solar eclipse : this will make the expressions for the direct images of occultations the same as those for the inverted images of eclipses of the sun, in estimating the contacts either from the north point or from the vertex. 390 SPHERIUAL ASTRONOMY. Parallactic Angles for the Latitude of Greenwich, (same sign as /*) Arguments : Apparent Hour Angle and Declination. Hour Angle h. Dec. North. o IO 20 3o 4o 5o 60 70 80 90 100 no 120 i3o 140 o o o o o o o o o o o o o O O 8 i5 22 27 3i 35 37 38 3 9 38 37 35 3i 27 I O 8 i5 22 27 32 35 37 38 3 9 38 3 7 34 3i 27 2 o 8 16 22 28 32 35 37 38 89 38 3 7 34 3i 27 3 o 8 16 22 28 32 35 37 38 3 9 38 36 34 3i 26 4 8 16 23 28 32 35 37 38 39 38 36 34 3i 26 5 o 9 16 23 28 33 36 38 39 39 38 36 34 3o 26 6 o 9 17 23 29 33 36 38 39 3 9 38 36 34 3o 26 7 9 17 24 29 33 36 38 3 9 3 9 38 36 34 3o 26 8 9 17 24 29 34 36 38 39 3 9 38 36 33 3o 25 9 9 17 24 3o 34 37 38 39 3 9 38 36 33 3o 25 10 o 9 18 25 3o 34 37 39 39 39 38 36 33 3o 25 ii 9 18 25 3i 3S 37 39 39 3 9 38 36 33 2 9 25 12 10 18 25 3i 35 38 39 4o 3 9 38 36 33 2 9 25 i3 10 F 9 26 3i 35 38 39 4o 3 9 38 36 33 2 9 25 i4 10 J 9 26 32 36 38 4o 40 3 9 38 36 33 2 9 25 i5 o JO 1 9 27 32 36 3 9 40 4o 3 9 38 36 33 2 9 24 16 o II 20 27 32 37 39 4o 4o 4o 38 36 33 2 9 24 *7 o 1 1 20 28 33 37 39 4o 4i 4o 38 36 33 20 24 18 o II 21 28 34 38 4o 4i 4i 4o 38 36 33 2 9 24 '9 o 11 21 29 34 38 4o 4i 4i 4o 38 36 33 2 9 24 20 o 12 22 29 35 3 9 4i 4.i 4i 4o 38 36 33 2 9 24 21 12 22 3o 36 39 4i 42 42 4o 3 9 36 33 2 9 24 22 12 23 3o 36 4o 42 42 42 4i 3 9 36 33 2 9 24 23 13 23 3l 37 4o 42 43 42 4i 39 36 33 2 .9 24 24 r3 24 32 38 4i 43 43 42 4i 39 36 33 2 9 24 25 o i4 25 j 33 38 42 43 43 43 4i 39 36 33 2 9 24 26 i4 36 i 34 3 9 42 44 44 43 42 39 36 33 2 9 24 27 i4 26 35 4o 43 44 44 43 42 39 36 33 2 9 24 28 o i5 27 35 4i 43 45 45 44 42 4o 3 7 33 2 9 24 2 9 o if 28 36 4i 44 45 45 44 42 4o 3 7 33 2 9 24 By subtracting the parallactic angle, for the respective times of beginning, greatest phase, and ending, from the foregoing angles of position of the moon'.s centre from the north towards the eas x ,, wo shall evidently obtain the same angle* from the zenith or vertex towards the east. If, however, the operation be repeated for the accurate determination of the times of .beginning and ending, we shall have in the calculations the angle Jfalso at thf-se times. Let j, t*i, Mi be the angles appertaining to the beginning, and 12, i2, J/2 those for the ending, and we shall evidently have the following values, which will be more accurate than the preceding : APPENDIX XI. 391 Parallactic Angles for the Latitude of Greenwich. (game sign as Ji) Arguments : -Apparent If our Angle and Declination. Hour Angle h. Dec. South. o 10 20 3o 4o 5o 60 70 80 90 100 110 120 i3o 140 o 8 i5 22 27 3i 35 37 38 3 9 38 37 35 3i 27 i 8 i5 21 27 3i 34 3 7 38 3 9 38 37 35 32 27 2 o 8 i5 21 27 3i 34 37 38 3 9 38 37 35 32 28 3 o 8 i5 21 26 3i 34 36 38 39 38 3 7 35 32 28 4 o 7 i5 21 26 3t 34 36 38 3 9 38 3 7 35 32 28 5 o 7 i5 21 26 3o 34 36 38 3 9 39 38 36 33 28 6 7 14 20 26 3o 34 36 38 39 39 38 36 33 29 7 o 7 4 2O 26 3o 34 36 38 39 39 38 36 33 29 8 o 7 14 2O 25 3o 33 36 38 3 9 39 38 36 34 29 9 o 7 M 2O 25 3o 33 36 38 3 9 39 38 37 34 3o 10 7 M 2O 25 3o 33 36 38 39 3 9 39 37 34 3o ii o 7 M 20 25 29 33 36 38 3 9 3 9 39 37 35 3i 12 o 7 U 20 25 29 33 36 38 3 9 4o 39 38 35 3i i3 7 '4 J 9 25 29 33 36 38 39 4o 39 38 35 3i i4 7 i3 *9 25 29 33 36 38 39 4o 4o 38 36 32 i5 o 7 i3 J 9 24 29 33 36 38 3 9 4o 4o 39 36 32 16 7 i3 '9 24 29 33 36 38 4o 4o 4o 39 37 32 17 o 7 i3 r 9 24 29 33 36 38 4o 4i 4o 39 37' 33 18 o 7 t3 '9 24 29 33 36 38 4o 4i 4i 4o 38 34 T 9 o 7 i3 ! 9 24 29 33 36 38 4o 41 4i 4o 38 34 20 o 7 i3 '9 24 29 33 36 38 4o 4i 14i 4i 39 35 21 o 6 i3 '9 24 29 33 36 39 4o 42 42 4i 39 36 22 o 6 i3 X 9 24 29 33 36 3 9 4i 42 42 42 4o 36 23 o 6 i3 18 24 29 33 '36 39 4i 42 43 42 4o 37 24 o 6 i3 18 24 29 '33 36 39 4i 42 43 43 4i 38 25 o 6 i3 18 24 29 33 36 39 4i 43 43 43 42 38 26 o 6 i3 18 24 29 33 36 39 42 43 44 44 42 3 9 27 6 i3 18 24 29 33 36 39 42 43 44 44 43 4o 28 o 6 12 18 24 29 33 37 4o 42 44 45 45 43 4i 29 6 12 18 24 29 33 37 4o 42 44 45 45 44 4i ( beginning atest ph ending For < greatest phase f / of J) 's centre from K towards E. = -j ( ) ;<-) .-jr.) of D 's centre from vertex towards E. = < ( i ) M (20) These angles relate to the natural appearance or direct images of the bodies. For the same angles, as they will appear through ar. inverting telescope, 180 must be applied : this may be simply done by using (180 i) instead of ( i). SPHERICAL ASTRONOMY. To find the time when the apparent conjunction takes place, let t denote the interval, in units of an hour, to be applied to the time of the true conjunction, and h the common hour angle of the bodies at the true conjunction. Then the position of the sun, not being supposed to be influenced by parallax, the common apparent hour angle of the bodies, at the time of the apparent conjunction, will be h 1 = h -f 15 . t and therefore at this time, sin (* + 15. t), so that the conditi n for apparent conjunction, viz. a' = a A a = 0, gives for the determination of the interval t, which from this equation will be best found perhaps, by the usual method of double position. We only want, however, an ap- proximate value, and may therefore avoid much unnecessary labor in estimating this time. Thus, at the time of true conjunction, the same approximate formulae may be adopted as used at page 385, viz. _,, cos I , A a = P - - sin ft, cos D A*,=P'^sinl")^}cosA, \dt / cos D in which applies to the moon. It is evident, then, as the true positions of the bodies have no difference of right ascension, that A a is the apparent difference of right ascension ; and consequently, as the relative apparent motion in right as- cension is ai A s A, the correction t to be \d t / cos D applied to the time of true conjunction to get that of the apparent, will be rv COS I P' sin h cos D sm h t = /dh \ coal cos D /dh . \ t P' I sin 1") cos h ai - D r : (77 sin 1") cos h \dt / cos D P cos I \dt / the calculation of this expression, we mi\y us as a mean value of D. Assume, therefore, 100 cos D _100 cos U _ [0.23103] ' ~ P 1 cos I '~ ~57~ " cos / cos I To facilitate the calculation of this expression, we may use 57' as a mean value for P' and 14 as a mean value of D. Assume, therefore, 6 = 100 (^ sin 1") cos h = [1.40274] cos h gO= 100 sin A for which the nearest whole numbers will suffice, and we shall have The values of the factor/ are given for various principal places in the table at page 406 : for any place not contained in that table it can be computed from the above expression, and used as a constant factor for all eclipses at that place. The Talues of tf, 6( l \ are also tabulated at page 4')5, where, f >r convenience, the argu- zuent h Is given in time. APPENDIX XI. 393 II. FORMULAE OF REDUCTION TO DIFFERENT PLACES. Before quil ing this subject we shall give a method of calculating numerical equations which will serv.e to determine, with much ease and with sufficient accu- racy, the circumstances of an eclipse of the sun for any place comprised within a certain range of country. To effect this purpose in the most ample manner, in again proceeding with the general determination of the time of a phase, whose apparent distance of centres is A', we shall, in the expressions, separate as much as possible the quantities whi-ch involve the position of the place on the earth. The values of the co-ordinates x, y, given at p. 387, observing that a A a =', may be put down as follows: x = [(D -f a' corr.) - = ^ (3) Make now the following assumptions : (D + a' corr.) $ a cos D 1 ~ . A' A' k k = ; [(D -f a' corr.) 6] sin t -\ 7 a cos D' cos A A (4) AD A a cos D' . ] k k A q = ; A D sin i -\ ; A a cos D' cos t A A and> observing the above values of a- and y, the equations (2), (3) will become -<,- (6) 394 SPHERICAL ASTRONOMY. Let y, \L be determined by the equations ._ , _ corr) a' y COS ^ = ^ ; ' a COS D' y Sin w = ; A and/>, q will take the following values: p = y cos O// -f t) ) (7) It yet remains to determine the values of A p, A g, which depend on the po- sition of the place of observation. Adopting the notation used in the equations (3), (4), (9), (10), pages 379 and 382, we shall have [5.31439] A cos I . A a = - J . 7: sin A, 1 n cos U [5.31439] A \ . . n cos (A + i A a)l A D = fc I sm J cos D cos / sm D s \ I. 1 n! L cos i A a J To simplify the expressions, let _ [5.31439] A cos D' ~~ (I w) A' ' cOsZ>' [5.31439] A [5.31439] A c = L J_ . cos I>, a = 7T "" am 2) (1 HI) A' (1 ftnd b A' cos / sin h A a = j^ , cos D cos (A + i A ) A I> = c A ' sin / a A' cos Z cos i A a = c A ' sin / a A ' cos cos h -f- a A' tan - cos / sin A. These substituted in (5) give A p = c cos i sin / cos l\ a cos cos A ( a cos t tan 6 sin t) sin A I A q = A; c sin t sin Z cos / I A; a sin t cos A (k a sin i tan \-kb cos i) sin A I The value of b contains the factor =-. for which we have cos D --^?- = cos A D (1 + tan D tan A D). cos JJ Substitute the first value of tan A D t p. 379, and ' 1 p sin P ^ cos (A) cos D' cos D ' v ' = cos A D . 1 p sin P [sin / sin D + cos / cos D cos (A)]' Or, putting A instead of (A) in the numerator, which cannot sensibly affect the value of the fraction, cos D' 1 n f- cos A D . . cos D 1 H! APPENDIX XI. 395 This, supnosing cos A D = 1, reduces the values of the constants a, b, c, to the following : [5.31439] A 1 -(i^Srz 7 I . . (9) If e be a small arc determined by g cos e = b, g sin e = a tan - , we shall have a cos i tan -- b sin i = // H) = 6 cos x * i" = fc c sin t \ y" co&W H) = kasint 1. ....... (14) y" sin (//' ^T) =/fc6sinx ' and we shall have Ap = L' sinl y f cos J cos (V +h H)=. L' sin / y' cos / cos ty' + A), &q = L" sin / y" cos / cos ($" + h H)=zL" sin I y" cos / cos (<" + A)> so that the equations (6) will become cos u = p L f sin ^ + y' cos / cos (^' + X) = (T 7 - ? ) T Ar sin u) -f- i" sin / - y" cos J cos ty" + X) ' After computing the constants k, p, q, L' t L", //', ^", by means of the equations (1). (H ( 8 ) (9), (10), (11), (13), and (14). we shall thus have two numerical equa- tions for the determination of o> and the Greenwich time t of the phase, for any place whose latitude is / and longitude X. The accuracy of the determination will principally depend on the proximity of the resulting time t to the assumed time T; and therefore the result will be near the truth for all places where the phase will take place near to this time. In making these calculations for any particular portion of country, which for the partial |>hase will be necessary for lioth the beginning hd ending, it will be best in the first instance to fix upon a place near the centre ui.d compute the eclipse for that place, which computation will furnish good .mean values for the data D t t, ft, a' corr, A D, A a, i, yi, A', A, and comp. log (1 ,). 396 SPHERICAL ASTRONOMY. f cos V - y', f cos I' 1 = y" By suppo^g ' ^ f I * ' x , J, s . u , = I L . . . . (It) the expressions U Bin I -f- y' cos I cos ((//' -f" A), L" sin J -f y" cos / cos (t//" -f X), will take the forms (' [sin /' sin / -f- cos /' cos I cos (V -f A)], {" [sin f" sin I + cos J" cos / cos ( cos t X' = b sin /' ; y' - 6 cos /' (18) ein V = cos J) sin c = A 6 sin I" ; y" = ko c-us t i ^ /j^-v U (^-^) = ^L, v * ' cos I" APPENDIX XI. 397 These may be employed instead of the equations (13) and (14) ; or the equations (18) and (14) may be adopted in their reduced form, viz. : = cos D cos i o j- cos (i// H)=. sin D cos c ..,,. (20) - sin (if/' H) = sin t 6 T " - = cos D sin k b ^7 cos (;//" H) = sin D sin (21) j-r sin ($" H) = cos i in which the coefficients c, a, will not be required. III. TRANSITS OF MERCURY AND VENUS OVER THE DISK OF THE SUN. These phenomena are, in many respects, analogous to that of an annular eclipse of the sun, and admit of a similar calculation ; the principal distinction consists in the negative sign of the relative motion of the planet in right ascension, which will make the inclination of the orbit always obtuse, and therefore render some modifications necessary in the determination of the particular species of the other angles which enter into the computation. To avoid any confusion that might thus arise, we shall adopt the sun as the movable body, and refer his positions to that of the planet which we now suppose to be stationary. Thus, 6 = the O's declination; ]) = the planet's declination ; it = the O's equatorial horizontal parallax, P = the planet's equatorial horizontal parallax; = O's right ascension minus that of the planet; x (V -f a' corr.) D ; y = a' cos i' ; a-j = the. O's motion in declination minus that of the planet ; y\ = (O's motion in right ascension minus that of planet) . cos 6' ; and so we might proceed as with nn eclipse of the sun, only observing that the relative parallax p ( P) is a negative quantity, and that the positions of the contacts on the limb of the sun, as in the case of an occultation, will be at points opposite to those which come out in the calculation. However, as the relative parallax is always very small, the ingress and egress of the planet will be seen at all places on the earth at nearly the same absolute time ; it will, for this reason, be best to compute first the circumstances for the centre of the earth, and then to ascertain the small variations produced by parallax for any assumed place on the surface, which may be readily deduced from the preceding equations for the reduc- tion of an eclipse of the sun. Let w, (t), be the values of , t, for the centre of the earth, and, by separating the effects of parallax from the equations (6), 398 SPHERICAL ASTRONOMY. .. cos w = p, A cos w = A p, A t = A q T k A sin w. But, as the quantities A cos w, A sin w are very small, A sin w = A cos w - - sin w that is, A sin w = A - - . Therefore, r sin w cos w /, cos w \ A t = A q k Ap -. - = (k Ap- - T A q). sin w \ sin w / In this expression substitute the values of A p, A q, according to the equations (12), and we find A t = .r. cosf iTw].. / cos[ tTw] , T co8[~ xTwl . 7 \1 1 ke - K --- sin I cos ilk a - - J cosh-kb - L . J sin A) |, L sin w \ sin w sin w / J ?-&- -, c = 6 ccs 5 and a = 6 sin a. in which b = A A Because of the smallness of the parallax, the angle ewill not be appreciable, ami consequently % = 90 -f *, cos [ x T w] = sin [ t qp w]. We shall therefore have for the time of ingress or egress the following general expression, in which the terms within the brackets depend on the position of the place of observation ; also the upper signs apply to the ingress, and the UT^aer signs to the egress. / = T q T k sin w .cosf Tw] . /. cosf tTw] 8in[--iqFTr] . .\ ,1 cos<5 - ^ -- -sin/ |sm<$ =; --- -cosh -- ^ - -- isuaAlcog/ I sin w \ sin w sin ,v / J Assuming k" = : - , this expression will resolve into the following : p sin w A 08 ' (a -f corr.) D y cos $ = ^- '- A y sin i// = COS W =: y COS v = k . L" (U/ -j" *) -} (*) <<) A sin w ~ = cos[(-0 T w] cos * y" cos &" H) = cos [ (- i) T w] sin t (<*) APPENDIX XI. 399 In these equations, H= the Q's true hour angle from the meridian of Greenwich, at the time (t). For \ extei : ior I contact of limbs, A = \ ' + S I ( interior } ( a s } For contact of centre of planet with 's limb, A = a ', s denoting the true semi-diameter of the planet, and which will be sufficiently accurate for all places on the earth. The time Tma.y be assumed near to the time of conjunction in longitude, or right ascension, as it may suit convenience. For Mercury, if very minute accuracy is wanted, it may be necessary, for more correct values of (t), to assume two times T near to the times of ingress and egress ; but it is very questionable whether such a precarious extent of accuracy would sufficiently recompense the time ex- pended on the calculation. IV. OCCTJLTATIONS OF STARS BY THE MOON. These may be calculated in the same manner as eclipses of the sun, the only difference in the operation consisting in the star having neither motion, parallax, nor semi-diameter. But where great minuteness is not wanted, these particular circumstances will afford some degree of simplification to the expressions, if that parallax of the moon be adopted which would answer to the star as an apparent place, since this parallax, at the times of immersion and emersion, will then be precisely that of the respective points of the moon's limb which come in contact with the star; and thus the augmentation of the moon's Sfmi-dinmeter will be evaded, so that the true semi-diameter may be employed. For this novel and ju- dicious expedient we are indebted to Carlini. See Zach's Correspondance, vol. xviii., pnge 528. As in the case of the pun, let & denote the declination, and h the hour angle of the star, and let, P represent the equatorial horizontal parallax of the moon. Then, for the effects of parallax in right ascension and declination, we must sub- stitute S for D' t and h for h in the formulae (2) at p. 379, which thus become, dis- regarding A A, ,, cos I . A o = p P = sin h, 008 D A D = p P (sin I cos S cos I sin & cos 7i). As soon as the immersion takes place, these expressions will represent the parallax of that point of the moon's limb which is in contact with the star; and therefore the application of this parallax to the centre of the moon will produce nn apparent distance A' of the centres, equal to the true semi-diameter s of the moon. Also as the star, in the course of the occultation, is only affected with its apparent diur- nal motion, the hourly variations of the above values will be 400 dh. SPHEE-ICAL ASTRONOMY. _ /dh . ,,\ cos I A ai = n P I - sin 1 I cos h. \dt / cos D A Di=pP ( sin 1" 1 cos / sin 6 sin 7* ; in which is 15 2' 28", the hourly diurnal motion of the earth, and therefor* sin 1" = [9.41916]. Assume 7 . cos T = peos/= , , (2) = p sin 1 = (1 cos 5 0O sin 3 cos /<) . P, ^ J) 1 =. 0( 3 ) . P sin J sin A. If, in the values of A a, A ai, we use cos 6 instead of cos D, the values of x, y, x lt yi, p. 387, will become x = (D 3) (0( 2 > . P cos t 00) . P sin <5 cos A) y =a cos 6 0' 1 ' . P sin h %i = Di 0t 3 ) . P sin 6 sin A yj = cti COS 5 0O . P COS h in which we have disregarded the a correction. With the values of *, y, xi, yi, so found, we may then proceed with the equa- tions (16) and (18), pages 387 and 388, as in the case of a solar eclipse. This method is similar, and, as far as accuracy goes, the same as the recent method of Professor Bessel, who divides all the quantities by the equatorial hori- zontal parallax of the moon. He assumes P = cos (3) u = 00) sin h, u' = 0( 3 ) cos h ) v = 0' 2 ) cos & 00) sin 3 cos h, v' = 0' 3) sin 5 sin h ) so that if we change the signification of the symbols x, y, x i} yi, and suppose them now to represent the preceding values divided by P, we shall have x=.q v, Xi = q' v' v = n u, t/i = p' U (*) (6) Tbeae values being adopted, in proceeding with the equations (16) and (18) we must use A' == , the value of which, according t r Burckhardt's Tables de la Lune (Paris, 1812), p. 73, is [9.43637]. Much facility is thus given to the calculation of occultations, for different plnces, if the values of p, q, p', q', which are indepen APPENDIX XI. 401 dent of geographical position, are published; but if these quantities are to be pre- pared by the computer, the equations (2) will be more simple and advantageous. The chief difficulty in the calculation of occulta- tion?, for any particular place, rests in the selection of the list of stars : in the course of any year a great number will be liable to occupation on the earth generally, though the majority of them will not be occulted at the particular place for which the special calculations are to be made. It will therefore be expedient to reject such stars as may at different stages of the calculation be shown to violate any conditions necessary for the existence of the occulta- tion, its appearance above the horizon, or its exemp- tion from the glare of sun-light. For the general list we may observe, that the difference of declina- tion at the time of conjunction must be within the limit of about 1 30', and that all stars, whose con- junctions with the rnoon occur within two days of new moon, may be omitted. In the process of exclu- sion for the particular place, the first and most pal- pable condition is, that at the time of conjunction the sun must be below, or near to, the horizon ; if more than half an hour above the horizon, the occul- tation will surely be useless; another condition is, that the star must be above the horizon; and, to satisfy this, the hour angles at the times of immer- sion and emersion must be less than its semi-diurnal arc. The value of the hour angle at the time of apparent conjunction may be determined by increas- ing that at the time of true conjunction by the qnan gd) tity , according to the tables on pages 401 a, .J 6 and 402 ; and it may be observed that this hour angle must not exceed the semi-diurnal arc by more than half an hour. For the latitude of Greenwich, the semi-diurnal arcs, allowing 33' for refraction in the horizon, are shown in the annexed table. As a final test for the exclusion of unnecessary stars, it is useful to calculate the extreme limits of latitude between which the star will be visibly oc- culted on the earth. These will evidently appertain to the extreme northern and southern points of the northern and southern limits of contact, determined as for a solar eclipse, a point in the northern or southern limit will depend on the formula} Nos. 27, 28, pages 359-60. Thus, Dec. of Star. Semi-diurnal Arcs, for the Latitude of Greenwich. Dec. North. Dec. South. h. m. h. m. o 6 4 6 4 6 9 +5 2 6 i4 * 5 54 ^ 3 4 6 19 I 6 24 549 ; 5 43 6 5 6 29 5 5 38 5 6 6 34 533 5 7 6 3 9 ^ 528 5 8 6 44 523 5 9 65o 6 5,8 5 10 655 5 5i3 5 5 6 1 1 7 o ^ 57 12 r 6 7 6 5 2 : i3 5 7 " 456 6 fi 5 i4 7 17 4 5l 6 v5 23 6 4 45 5 16 17 7 28 7 34 4 4o A 434 ; 18 7 4o 4 28 '9 747 7 4 22 20 7 53 4 l5 I 21 8 o 7 4 9 ! 22 8 6 6 4 2 23 8 i3 7 356 6 24 8 21 3 4 9 ' 25 8 28 34i 8 26 8 36 334 ' 27 8 44 3 26 ' 28 8 53 9 3 18 J 29 9 ' 3 9 9 3o 9 V 3o-9 cos w = n A' M= * and thence, sin / = sin D' cos Z + cos D' sin Z cos M. 20 402 SPHEEICAL ASTRONOMY. It is now our object to ascertain what value of ' will render the value of /, so deduced, a maximum or a minimum, and what will be the corresponding value of I. Let be an arc determined by the equation, cos Z = cos sin w .......... (6) Then by uniting with it the equation cos <>' sin Z = cos w . . . . . . (7) we infer that sin ' sin Z = sin sin w . ..... . (8) because the squares of these three equations added together will give unity on each side. By these equations we shall hence have sin D' cos Z = sin D' cos sin w, sin Z cos M'= sin Z (cos i cos ' T sin i sin w'), = (cos a/ sin Z) cos t T (sin ' sin Z} sin i, = cos i cos w T sin t sin sin w ; and, consequently, sin I = cos D' cos t cos w -}- sin w (sin D' cos T cos D' sin i sin 0), which now involves only one variable . Again, assume two arcs, 6, \}/, which will fulfil the equations, cos 6 cos i// = sin J)' .......... (9) cos 6 sin i// = cos D' sin i ........ (10) A. third equation will follow from these, viz. : sin 6 = cos D' cos * ........... (11) because, as before, the squares of these three equations will together make unity. The value of sin / will now become sin I = cos w sin 9 -{- sin w cos cos (0 -f" t//). The angle + \f/ being the only variable in this expression, it is evident that the greatest value of I will have -|- \f/ = 0, and the least -f- \f/ = 180. Therefore, These would be the extreme latitudes for the appearance of the occultation if tl e earth were a transparent body ; as this, however, is not the case, it will be nect-s- sary that the star should be above the horizon, a condition not included in the preceding equations. The zenith distance Z must not exceed 90, and therefore cos Z must necessarily be a positive quantity. By the equation (6) cos Z must have the same sign as cos 0, and this must be the same as -|- cos i// for northern limit, or cos \l for southern limit, because in 'the former case -j- t// = 0, and in the latter ^ -j- !// = 180. But, by (9), cos 's right ascension, to the tenth of a minute, In are. K zzzz n = k cos A, A a = [5.31439] k sin A [corr. for M], A 01 = Qi n, A Di = Q 2 sin A. Correction for n to be taken from the table on page 383. (A) = A -f i A a, tan 6 = cos (A) cot /, G = cos (A) cos /, sin " tan M i^. cos (e -f- Z>) tan (A), tan e = tan (6 -f- D) cos Jf f B = cos Jf cos t ; sin S # J/ to be in the same semicircle with A. APPENDIX XI. 4-07 M, = A sin e, A D = [5 31439] A B [corr. for i], s' = s [corr. for ni\. Fo ( partial J ^ A , = U + .) ( total or annular ) ( ' ~ o ) Correction for Wj to be taken from the table on page 383. D' = D A D, a' = a A a, y = (a A a) COS Z>', y : = ( ai A i) COS J9', = (2)' -f- a' corr.) S, x, = A A Di. sin S cos /S' TFcos* [8.55630 TP cos [ (S -f )], ^ = -- - - n IT 7. COS W = - :, C = A ' cos ( = e sin a, < 2 = c sin b. ending ) Time of greatest phase = \ sum of times of beginning and ending. When n < ' ~ , the eclipse will be total if s' > , or annular if ' < v. in thia ease these last equations No. 7 must be repeated for this phase with A' = d ~ * the results of which ought to give the same time for the greatest phase. Take A ' for partial phase, and Portion of sun's disk eclipsed = A ' n. Magnitude of eclipse = , the sun's diameter being unity. 8. For the positions of the points of contact on the limb of the sun, At | be ^^ | , angle from north towards east = | < ( ~ | j ~ ^ for direct image. At \ be ^'" nin g 1 , angle from north towards east = j (0 - ) - . ) for inverted i ending ) ( (180 1)+) image, For the position of the moon's centre at greatest phase, Angle from | "^^ I towards east = || "~ || I for direct image. Angle from | ^ J toward, east = | ||^ - 0_ ^ | for inverted [m ^ 408 SrrfERICAL ASTRONOMY. 9. For a more accurate calculation of the time, ) + <** I f or <# mi image. ( vertex ) { ( , a ) + U2 Jfa ) Angle from J north i towards the east = | ( 18 " ~ "> + W2 I for inverted ( vertex ) ((180 2 )-f-<>2 Jf a J image. II. - FOKMUL^E FOB REDUCTION TO DIFFERENT PLACES. 11 Instead of Nos. 5, 6, 7, substitute the following: D' = D A D, ' = a A a, a?, = Z>! A A, y! = (a, A o!) cos D', tan = -, k = [3.55630] A ' cos ' ' corr.) i a cos D' y cos ^ = - - --y ^ - , y sm ^ = - -^ , = y cos (^ -f ), ? = ^X sin (< + i), [5.31439] A r 12. 6 = = - r- J fcorr. for nil A e in minutes = [7.9208] A a sin D, % = (90 + ) #. 18. ^Tr= the true Greenwich hour angle of J) at the time T. jj jj< -r- = COS D COS I. 7T = COS D sin t to rcos (0/ ^) = sin D cos , -^ r cos fy" -ff) = sin D sin , c - wn (^' T) = cos x, rr sin <^" //) = in X- APPENDIX XL 400 14. The constants T ', k, p, IS, L", y', y", being so computed, the angle u and the time t of the phase for any place whose north latitude is / and east longitude X, will be determined by the two following equations, in which the upper sign re lates to the beginning and the under sign to the ending. cos w = p L' sin I -f- y' cos I cos (A -f- *. For contact of planet's centre with sun's limb, A = * A For | exter ? or I contact of limbs, A = \ ff + ( interior ) ( a tan t ! cos & cos t cos <5 A (6 -f- a corr.) D a cos y COS t// = ' , y sin i// = , A A cos w = y cos (t/> -f- )> y = A; y sin ( sin l\ osinp' the uppe- signs for the ingress, and the under signs for the egress. The positions of the points of ingress and egre. These angles may be assumed to be the same for any place on the surface, the effect of parallax being so very minute. IV. - OCCULTATION OF A STAR BY THE MOON. GENERAL LIMITS OF LATITUDE. 1& (ai and Di at true 6 ) tan i = , n = (diff. dec.) cos . i cos 6 cos Wj = =F -p .2725, cos w 2 = =F -f .2725, sin = cos i cos t, li = Wi Q, sin / 2 = =p cos & cos (wa )> w^ Ws, i, 0, same sign as 6, v- 555S When Wi is impossible, l\ = 90, with the same name as 3. When Wa is impossible, /a = complement of <5, with different name from i. CALCULATION FOR PARTICULAR PLACE. 19. For the latitude of the place prepare the constants dd) ?(') = p cos /, $ : = f sin / = -, f () = [9.41916] *(>, rhich will serve for all occultations at that place. For the time of true <5 find h = sidereal time at place right ascension of star ; and thence determine the time T, as in No. 1. For this time take out the quanti- ties P, s, D, Di, a, ai ; and compute x = (D 6) ($W . P cos a f'V . P sin 6 cos h) ; y =s a cos 3 0' 1 ) P sin A ; ^ = D! #(3) . P sin 5 sin /< ; ^! =S Oj COS ^ - 0( 3 ) . P COS A. With these proceed as in Nos. 6 and 7, using A ' = s = [9.48587] P. 20. For the positions of the points of immersion and emersion on the limb of the moon, At ( immersion j | , angle from north toward8 east= j (180-)- ) for ^ . < emersion )' ( (l80-) + w ) At \ Immer810n j. angle from north towards east = ! (')- I f or invr^ image. ( emersion ) ( ( ) + <" ) For the same angles from the vertex we must deduct the parallactic angle for each time. 21. If an accurate calculation is wanted, proceed as with a solar eclipse. APPENDIX XI. ill V. - ECLIPSE OF THE MOON. i2. Fix on a convenient time near to the time of opposition in longitude, or full oon ; and for this time find P, s, , ~ w I for direct image. < emersion ) ( (180 ) -f w ) At \ iramersion I , angle from N. towards E. = \ ^ l \ ~ w [ for inverted image. ( emersion ) (( i) -f w J At the middle of the eclipse, / cent, shadow from N. towards E. =J ( 180 ~ f) [ for ] dim * , J image. ( ( t) ) ( inverted } To get the same angles from the vertex, the parallactic angle must be deducted for the respective times. J3 & 5*-O 412 SPHERICAL ASTRONOMY. Examples. I. - ECLIPSE OF THE SUN. Let it be required to calculate the circumstances of the solar eclipse of May 1ft 183G, as it will be seen at the observatory of Edinburgh. The elements of this eclipse are stated at page 362. h. m. s. Greenwich sidereal time at Greenwich ) mean noon Longitude ........ 12 43-6 W. Edinburgh sidereal time at Greenwich ) mean noon ....... [ 3 2O '*4 " ' *~ '? Sun's right ascension at d . . . . 3 29 25-2 f . 3 o3 Hour angle h at Greenwich mean noon o 910-8 83 i ( Greenwich mean time of 6 .. 2 21 22-9 -8 I Acceleration ....... 23 2 aj ./ 84 . Aatc5 ... + 2~73 '. . 6+21 SO + 55 (4- -87 a ,./_g+ 63 5o-4 T6 h. m. Greenwich mean time of true c5 . 2 21 g ... 4- 19 33 43 a + A D . . f 35 26 L a + i it 23 49 . '... + 18 58 17 j a' + a' corr. . o | log + i S . . . 4- 16* 58 29 cos D' + 9- o 34 ( 53i48 \ log . + 97574 . . . . + + 22 I 3 . i 2090 9.97574 O 12 1.50722 y i + 3-09664 (I) SPHERICAL ASTRONOMY. ' / 8 . . + no 28-0 . -j i . . + 19 53-5 y - * . tan . cos S . W . . cos . ft log A' . COS W . + 1.50722 I -07918 yi . + 3-09664 (i, a? . + 2-655i4 0*42804 cot i + o-445o 9 .54364 cos t + 9-97328 . . + 1-53554 const. 3-5563o + 5^>65iT (2) + 1-53554 9.81129 (S + ) i3o 21-5 . Partial . . w + 90 4ii 1-34683 3-26677 //+ i- 9 6848 (2) (i) . . 8-08006 8 - 08006 a 221 2-6 c . b 89 4o-4 sin a ti Assumed time. Beginning .... Longitude .... PARTIAL. Beginning .... n . Annular . . log A' . + n533'-9 cosw . - (S + ) i3o 21 .5 c . 3-88842 + 9.81732 c 3-88842 sin 6 9-8o5io Greenwich mean times V. Edinburgh mean times Greenwich mean times. V. Edinburgh mean times. 3- 7 o5 7 4 + 3- 6 9 352 h. m. 8. i 24 39 3 i3 h. m. s. , + I 22 l8 . . . 3 i3 i 48 21 12 44 Ending 4 35 i8-j W. . . 12 44 ^ i 35 37 Ending 4 22 34 -j 1-34683 i -71181 H+ 1-96848 9-635o2 o-635o2 2.33346 + 9.9604? 2-29393 c 2-33346 sin b 9-40711 a 245 55 '4 sin a b 14 47 .6 tj Assumed time. Beginning .... Longitude . ANNULAR. Beginning .... + i -74057 h. m. s. o 3 17 3 i3 h. m. s. < 2 + o o 55 . . . 3 i3 3 9 43 12 44 Ending 3 i3 55 j W. . . 12 44 \ 2 56 5 9 Ending 3 i n ] POSITIONS OF CONTACTS FOR DIRECT IMAGE. Partial contact &t j beginning ending Annular contact at beginning ending . ( 19-9 *> + 90-7 no-6 ) ( west g c from north to wards 1 . o n5.6 i35-5 9 5 :l\ ( west from orth towards! APPENDIX XI. 415 For the same angles from vertex we must estimate them towards the east, and deduct the angle M, thus o o Beginning t355 Ending + 96-7 M + 3i-9 M + 3i- 9 1 67 4 towards west. 63-8 towards east COMPUTATION FOR i h 48 m . FOR AN ACCURATE DETERMINATION OF PARTIAL BEGINNING. D + 19 19 35-9 $ + i Di + 9 26 Edinburgh Sid. Time at Greenwich Sidereal Equivalent for -5 H 's R. A. 8 67 3 9 -3 a i5 23.2 ai + 27 38 h. m. s. i Mean Noon . 3 20 i44 . . 48 7.9 5 8 32.2 3 28 18-2 m . . 7-9^712 cos D . 9.97481 {time arc const. . + 26 3'. f 5.3i43 9 7.97231 + 9-62690 + 2.91730 + i3'46"-6 + 9.96666 + 9-83256 k . . cos h n . log . . h . . i A a . (70 . . e . . D . . e+D . M l . + 9.96707 7-92938 '4-7172 sin A . corr. for n . ( log . . . 1 A a . . COS . cot / . . tan 6 . . sin . . cos . tan (h) . . j tan M l \ cos Mi . . tan (6 + D) (tan . . J cos . . . sin . A ... n . #a . . . 2-1764 j log . . . + i-8o33 ( A Di . . + i' 4" + 9.96666 + 7' 23" + 25 3-5 + 6-9 + 26 10-4 + 3i 36. 7 + 19 19-6 cos I . . +9.76001 + 9.78922 a . . . +9.70667 B . . . + 9.78666 . . check . +9.92002 + 9.71946 + 9.79945 + 9-92001 + 9.67209 + 5o 56-3 t + 21 21 I ' ' + 48 55-9 + 9-69210 + 0-09068 cos e . . + 9.81764 + 0-06979 + 8-19711 B . . . +9.78666 const. . . 53i439 5 ioio4 * 8.19711 corr. for Wi 5i8 J- 8- o 7 444 j log . . . 3-3o333 . +33'3o"-6 416 D' a'c ^ X SPHERICAL ASTRONOMY log* . 2.94904 5i8 (log 2.95422 Di . . 4- 9 26" iv . . i5 o-o A D 1 ..4-i4 a i5 4 : 9 . 9 x, . , . 4- 8 22 A' . . 3o 49-9 n t a t a 4. 1919 35-9 a i5 23 '2 i . + 27 38 + 33 3o-6 A + i3 46-6 A i . + 7 a3 jjh 1846 5-3 ( o.' 29 9.8 4- 20 1 5 2-2 (log 3. 24299 log. - +. 3 .08458 i- 18 57 3 9 .3 cos D' . + 9' 97627 - + 9 .97627 . it 3i-8 y 3- 21926 2/i . 4- 3 06086 (r) x . r 2- 83 99 8 x l . . 4- a . 70070 O ' - 112 39.8l j tan S . (sin S . f 0- 9" 37928 96610 COt ii COS ! + + 9 .36~o75~ ^62^5 h 2 3 34.49 W. . 4- 3. 25416 . . + 3 .26416 \- 89 5-32 cos 4- 8- 20168 const. 3 -5563o / n . + 6 77261 (2) i 45584 log A'. + 3- 26716 H . . 4- 3 71176 (2) (i) f- 89 6-93 COS 0>i . + 8^ 18869 - - - + 8 18869 - o TTeT c . - + 5 62307 6-463 7 3 61 o-2o683 2-i 9 363 tl . . o h 2'"36 3 Assumed time i 48 o Beginning Long. . I 45 24GrecnhM.T. 12 44 W. 32 4oEdin. M.T PARTIAL. Beginning . If th^ -jalculation be repeated for the Greenwich time i h 45 rn , it will lead to ex tly the same result, which is therefore to the accurate second, according to the data employed. POSITION OF CONTACT FOR DIRKCT IMAGE. o ( 1,1 . . 23-6 112.7 4- 21.4 ( tl ) wi J/i . . i34-i The point of contact is therefore j j.jj ] from j - towards west APPENDIX XI. 417 II. EQUATIONS FOR REDUCTION OF PARTIAL BEGINNING. The data for this computation are taken from the preceding one. 3.5563o 5- 3 i 439 A' 3-26715 A . 8.19711 cost 9-96215 corr. for MI 5i8 7-9208 A a . +2-9173 sin D +9.5198 6-78560 3.5i668 + 0-3579 e + 2-3 y, 3.o6o85 A' 3-267r5 k +3.72475 6 . +0-24953 k . +3.72475 kb. +3.97428 ' D + 19 19 35.9) a corr. 2.2) 90 + t . n3 34-5 x . n3 32 2 a 2.96530 cosD' +9-97627 t + iS 57 39.3 A'y sin i// 2-94157 + 21 58-8 . . . A'y cos \Li +3- 1 20 1 8 U, 33 32-2 ( tan i// 9-82139 , + 23 34-5 {cos i// + 9-92092 A'y. . + 3*19926 A' . . 3.26715 cos (d + i) + 9.99341 j + 9.92552 Long. 3 10 .9 W. sin (^ + i) 9238o2 k . . 3.72475 j 2- 89488 ( q . . o h i3' n 5 s T . . + i 48 II + 28 :4 -4 cos D + 9.97481 cos i + 9.96215 6 . +0-24953 L . +0-18649 sin D +9.51977 cos + 9.96215 T . . + 2 i 5 cos D + 9-97481 sin t + 9-60200 kb . +3-97428 x" . +3.55109 sin D + 9-51977 sin i + 9-60200 + 9-48192 cos x 96oi34 + 9-12177 sin x +9-96228 , j tnn . 0.11942 V ~~ /sin . 9.90109 ^ (tan . + o-84o5i 1 4 ?" \,-- u . +9-9955, #+28 i4-4 +9.70025 tf+ 28 14 -4 + 9-96676 ^' . 24 32 4 6 . 0*24953 ^ y' . +9-94978 1 + TIO i -5 kb . 3.97428 y" . +3.94104 27 418 SPHERICAL ASTRONOMY We have hence, for the Greenwich time t of beginning, at any place whose lat- itude is I, = north, south, and longitude X, -f east, west, the two following equations, which may be safely depended on for any place in Scotland or the North of England. cos w =0-84240 [0-18649] sin J+ [9-94978] cos/ cos (X 243a'-4) f = 2 h i m 5 [3 .72476] sin w + [3- 55109] sin/ [3. 94104] cos /cos(A + 110 i'-5) Contact on 0's limb, w -f 23 34' -5 from the north towards the west. As a check on this calculation take the assumed radical place, Edinburgh, and / = + 55 46'-9, A= 3 io'-9, giving w = 89 6'- 9 and t = i 1 - 45 m 24 s , which perfectly coincide with the results of the original calculation. Similar calculations for the ending of the eclipse give the equations, cosw 0-93848 [0.20291] sin^ +[9- 88677] cos I cos (X+ 27 6'. 7) <=i h 38 m 33 8 +[3.6689o] ! *inu>+[3.35544]sin/ [3.90073] cos /cos (A + 1 53 3' -8) Contact on 0's limb, w 16 56' -2 from the north towards the east. Also by calculating with T = 3 h i3' n for the annular phase there will result cos w = 2 9. 66600 [i-75 1 59] sin/ +[1-46950] cos /cos (A + i42'4) f = i h 43 m 7 8 T [2- 1 4475] sin w + [3-45484] sin/- [3.9255o] cos/ cos (X + r3i55'.9) Contact on 0's limb, 19 53'- 5 T u from the north towards the east, the upper sign appertaining to the beginning and the under sign to the ending, If cos o) > i, the place will be without the Unfits, and the eclipse "will not be annular. By taking / = + 55 46' 9, X = 3 io'9, the results will exactly correspond with the special calculation. Note. The expression of cos w for the annular phase, as the appearance of this phase is comprised within narrow limits on the surface of the earth, will afford a very convenient and simple determination of the places which range in those lim- its as well as those which range in the central line ; and we may expect very ac- curate results throughout the portion of country originally taken into considera tion. Thus for the southern limit we must obviously have cos w = -J- i, for the central line cosw = o, and for the northern limit cos u = i; and hence the following conditions: ( + i } I southern limit. p L' sin / -f- / cos / cos (A -f //') = < O> for < central eclipse. ( i ) ( northern limit. By making the assumptions ri cos 2F = y' cos (A -f i//') ) f ris\nN' = L' J *hey will give f p + i ^ ( southern limit ^ ri cos (N' +- /) = < p > for < central eclipse > ....(*) ( p i ) ( northern limit ) If we therefore take any meridian whose east longitude is A, these two equa- tions (r), () will serve to determine the extreme latitudes /, on this meridian, be- tween which the eclipse will be annular as well as that where it will be central For the preceding eclipse, these equations will be ri cos N 1 = [i .46950] cos (X 4- i 4a' 4), w'sin ^' = [r. 7 5i59] ; f _ [1.45737]) r southern limit. ri cos (2V + 1) = ] [i -47226] > for < central eclipse. ( [z .48665] ) ( northern limit. APPENDIX XI. If we take, for example, the meridian of Edinburgh, and use A= 3 io'9, there will result, o ' Extreme southern point of annular appearance, N. 54 19-7 Point of central appearance, N. 55 20 -4 Extreme northern point of annular appearance, N. 56 21*7 which are geocentric latitudes. III. CALCULATION OF THE TRANSIT OF MERCURY, November 7, 1835. The conjunction in right ascension takes place at out 7 h 38" 1 ; take therefore T= 7 h 4o m , and we readily find from the ephemeris the following data: i 16 i5' 58"2 O I II I II J) l6 22 42 a + O 10.95 A 2 32-6 a, + 5 32.7 9 4-8 v 16 10.4 P 12.66 , 8.66 With these quantities, the calculation, for external contact of limbs, is as follows: 9 16 10-4 < 4-8 A 16 i52 4'Qo . . . o6owO A 2-98909 6 + 7-61297 a -f I* 03941 ai + 2-52205 cos + 9-98226 + 9.98226 COS i + I.02I67 jC08W + 9'97^8 h. m. . *" + i-6366i g o 25 8-3 cos^ + 9.98226 r+ Ul__ *"co 8 ^ +"i- 61888 sin w + 9-97278 T q -f 8 5 8-3 k + 3-99G43 I n w + 3*9(1921 . . . 2 35 1 5. 6 Mean time of j "J^ f> ^ J'7 | for ^ ^^ rf the eartju 420 SPhERICAL ASTRONOMY. ffin CONSTANTS FOR REDUCTION OF INGRESS. h. m. s. 5 29 52.7 Equa. 4- 1 6 10-0 {time + 5 46 2.7 arc + 863o'. 7 i+ 25 32 .3 w 69 55 -4 __!_ w _ 44 2 3 H io5 58 -3 cos + 9854io sin S 9 .44?33 y 19 27 '6 *'' cos i + i. 61888 L" + 1.47298 CONSTANTS FOR REDUCTION OF EGRESS. b. m. s. 10 40 23.9 Equa, + 16 9-2 sin 9.84477 . __ 9 .3 OI 43 j tan + 0.54334 \ sin 9.98290 -f 9-86187 k" + 1-63662 y" + 1.49849 {time + 10 56 33 . I arc -f 164 8' .3 - 1 + W + 95 27 7 - . cos 8 sin S 9 77854 . . 44733 . sin + 9' 9980* + 8 4258 7 . . . . + 8. 42587 a" H + 88 o j tan + i- 5 7 2r5 T " ' ( i ,. ?r 9 99904 H- 107 23 7 k" cos 6 + i 61888 k" + 9' 99818 63662 L" 0-59742 i-6348o The former part of the calculation repeated for the times 5 h 3o m and io h 4o m we shall find more accurate times of ingress and egress, for the centre of the earth, to be 5 h 29" 56 and io h 4o m 3i", which, however, still cannot be depended on within a few seconds More reliance can be placed in the amount of reduction for parallax. The times reduced for any place whose north latitude is I, and east longitude X, viz. : Ingress, Nov. 7 (1 5* 29"" 56 8 + [i -473o] p sin / - [t -4985] p cos I cos (A 19 28') Egress, " " 10 4o 3t +[0-5974] p sin I + [i -6348] p cos / cos (X - 107 24') will indicate, with considerable accuracy, the difference between the times at any two places. The positions of the contacts on the sun's limb, for an inverted image, will be ( ingress 44 23' i ( west. Contact at ] ^ ^ .... ,5 28 \ ^om the north towards the ] APPENDIX XI. 422 IV. OCCULTATION OF A STAR. On January 7, 1836, the star Leonis, whose right ascension is io h 2J m a6*'4 ami declination N. i4 58' 89", will be occulted by the moon. LIMITS OP LATITUDE. At the time of true c5 in right ascension, viz., i2 h I2 m 17", we have the follow- ing data : D + i5 33 2 t + i4 58 39 D &+ o 34 23 with which we proceed thus: Dt ii 47 . 2-84942 a, + 3o 4i . + 3-265o5 Di ii 47 a, + 30 4l P + 56 4 to <5 + 14059' o ' + 1 47 24 9- 5843 7 cos + 9.98498 tan 9.59939 w 2 + 107 18 + 21 4i !tan 909939 cos + 9-96813 const, nat cos nat. cos 2725 diff. dec. + 34' 23" P + 56' 4" I + -56 99 + 3-3i45o n + 3.28263 + 3.52686 86 37 63 5i 8424 2974 log. cos + 9-9681 (i) log. cos + 8-8833 (2) log. cos <5 + 9-9850 (3) log. cos + 9.9531 (i) + (3) 9.75577 + 83 33 , 4 1 4 log. sin li + 8-8683 (2) + (3) The star may therefore be occulted between the parallels of latitude N. 83 33' and S. 4 '4. The parallel of Greenwich is within these limits; and if the hour angle of the star be computed roughly for the meridian of Greenwich, the star will be found to be considerably elevated above the horizon. A special calculation for the observatory of Greenwich will consequently serve as an <-xumple of the i.-- curastances for a particular place. CALCULATION FOR GREENWICH OBSERVATORY. Constants f >, >, 0( 3) . P . . . . 9'999 l3 cos I . . . + 9.79610 f") . . . + 9.79523 .... cot I . . . + 9.90381 const. . 0' 2 > . . . + 9-89142 f') . . These will be constant for all occultations at Greenwich. h. in. s. Sidereal time at mean noon . 19 4 22.4 Star's right ascension . . 10 23 26*4 h at mean noon . Mean time of true (5 Acceleration .... h at true 6 + 9.79523 9.41916 + 9-21439 i5 19 4." . i5 '9 4- 12 12 T . . . . + 1 1 6 2 acceleration . + i 4y 4 35 ,.j time arc 4 ii 14. ft 62 48'. 7 With this and a, = 3o'7 we find, by the table at p. 405, T= n h 6 m . h at mean noon is put down negatively, in order to have more readily the othor values of h lees than I2 h or 180. SPHERICAL ASTRONOMY. P 56' 4" . +3.52686 . . . + 3.52686 . . . +3.52686 1P>. . . + 9-89142 cos h + 9-65983 sin h 9.94915 + 3-41828 + 3.18669 3 47' 01 cos a . . +9.98499 sin 3 . + 9.41236 + 2.59905 sin 6 . + 94i236 2^88237 '.- 1 7, (1) + 9.79523 ;3> - 4- 9-21439 + 48... + 2-39428 2-10276 + 38 3 2' 7" D i . . + 47-22 Di . 1 1 4a t . > . + 9 1 9 Xl . 9 35 im 33' 54 7 ' | ., . + 3o' 44" 3-3o835 { =3-265 7 6 cos J . . +9-98499 cos a . +9.98499 1 j 3-29334 j + 3.25075 1 32' 45" 1+ 29' 4i" JP sin h . 3.47601 P . . 3.52686 PCOS/4 +3.18669 . +9.79523 const. . 9.43537 ^,(3) + 9.21439 t 3.27124 A . 2*96223 f + 2.40108 1 3i' 7" ( I.99I23 x . . . +2- 7 474i tan S . . 9-24382 8 ^ f cos (S. . . -f 9'99343 , o ' 9 566 cot i . 20 366 cos t . 2-75967 0.42474 . , + 9.97128 ; \ir 4. 2753o8 . . . . W . +2- 7 53 9 8 J ) + 3o 33.2 3.55630. .:.. n . . + 2.68906 + 6-28i56 A' ... 2.96223 H . + 3.09715 COSu, . - +9.72683 . w . + 57 47 -o . cos w + 9.72683 . . . +3.37032 a wn a . . 9-66045 6 27 i3.& c . . + 88 20. 2 sin 6 . + 3-37032 + 9.99982 3-o3o77 + 3-37014 h m , + o h 39 m .i 7* ... ii 6 T n 6 Immersion 10 48 .1 . Acceleration i 8 8 T. mean noon 19 4 -4 Emersion . Acceleration S. T. mean noon ii 45 i mean timus. 2 -0 19 4 -4 Immersion 5 54 -3 Star's R. A. 10 23 -4 . Emersion . Star's R. A. . . 6 5 1 5 . sid. tinue^ 10 23-4 ' j!m.A . 4 29. i =67 "1 Parallactic/ 39. 7 j Em, /* = . . ( Parallactic Z 3 3i .9 = 53" -T~3~6^ ( 1^ . . . +20-6 u> . . . + 57 8 From| nol 'f l ~" 37 J|tothe ( vertex 4-2 D ) (-<) - ( north east. From -j + 20 -6 + 5 7 -8 + 78 -4) to{he ^ c + ii5 3 ) f APPENDIX XI. 423 These angles are for the inverted image ; and, being estimated towards the east, the negative values must be considered as towards the west. The declination of the star gives for the latitude of Greenwich a semi-diurnal arc of 7* a3 m ; as this exceeds the value of A both at immersion and emersion, the immersion and emer- sion will both occur above the horizon. V. CALCULATION OF THE ECLIPSE OF THE MOON, April 30, 1836. The opposition or full mocn takes place at I9 h 58 m . For the computation assume the time ao h o. 8 19" b. in. 8. 20 h b. m. s. 21" b. m. s. >'sR. A. . . 14 32 5i-35 . . 14 35 11-19 . . 14 37 3i-43 's B. A. -\ - i2 h i4 33 52-38 . . 14 34 1.91 . . 14 34 n-45 ( time i i o3 + i 9-28 + 3 19-98 ( space 1 5' i5" + 17' 19" + 5o' o" a = + '7 '9 . 32 4l " .,= + 3,' 38' , + 5o o i9 h 20>> 21* O ' " O ' " ' * ) 's dec. a cor. . . -i4 5 19) . " . of ?4 19 58) . . 14 34 32 ) 5J 0's dec. . . +i5 6 35 . . + i5 7 20 . . +i5 8 6 x . . . +~i T~i6 + 47~27 + 33^ + 6 '' I6 "_ I3 ' 55 H i P = + 47 21 3 *i x\ ^ i3' 5^ + 33 29 + 3-01662 ai + 3.29181 P = 60' 19" cos D + 9- 98627 . . +9-98627 y + 3- 00289 t/i + 3.27808 z + 3-45347 xi 2-92117 P 3- 55859 9.99929 ' -f' 9 {tan $+ 9.54942 3.55788 P' . 6o'i3"> . 9 ) cot i 0.35691 CO8I + 9.96163 23 43 .9 17+3-47916 . . +3.479'6 a . l5 53 + 0+ 4 i3 -2 . cos +9.99882 3.5563o 44 29 n + 3. 47798 + 6.99709 *V . 44 External . . A' 3-568o8 i +35 38 .7 cos -|- 9.90000 #+3.71901 . . +9.90990 45 1 3 SHADOW 8 1 6 26 \ -~ J-39 25 '5 e -1-3.80911 5i 9 sin a 9-71716 c + 3-8o9t i sin6+9-8o685 , j 61 39 external ( 28 47 internal -3-52627 + 3.6i5o6 , - ^-".o < 2 + i h 8'".8 Assumed time 2o h o ... 20 o Beginning 19 4 o Ending 2 1 8 -8 Greenw b mean times. 4:24 SPHERICAL ASTRONOMY. For the times at any other place, it will only be necessary to take into account the difference of longitude. The positions of the points of contact on the limb of the moon may be deter mined in the same manner as those of an occultation, and will here be unnecessary. As A' for internal contact with shadow is less than n, no internal contact cau take place, and therefore the eclipse is only partial. The contacts with the penumbra are to be determined in a similar manner from the same values of , //, and will also be unnecessary here. A' for external contact with shadow 61' 89" n . 5o 6 Eclipsed .... 1 1 33 which divided by 2 s = 3a' 5z", gives o-35i for the magnitude of the eclipse, the moon's diameter being unity. APPENDIX XII. EQUATION OF EQUAL ALTITUDES. Let P be the pole, Z the zenith, S' the place of the sun in the afternoon, S the place he would have occupied had his declination or polar distance P S remain- ed unchanged. Make I = latitude of place = 90 P Z ; x = declination of sun = 90 P S ; a = altitude of sun = 90 Z S ; P hour angle Z P S ; then in the triangle Z P S, sin a = sin / sin x -f- cos / cos x . cos P ... (a) Differentiating, supposing x and P alone to vary, we have d x . cos x . sin I = cos I . cos x . sin P . d P + cos / . cos P . sin x d .r, or d x . (cos a? . sir I cos / . cos P . sin x) = d P . sin P cos / cos x whence tan I tan # \ sin P ~~ tan /7 APPENDIX XIII. 425 Denote by <5 the change in declination from the next preceding to the next following noon or change in 48 hours, and by t the interval in hours between the epochs of equal altitudes in the morning and afternoon. Then 48 : S : : t : dx, whence also which substituted in Eq. (6) give 6 . tan I . t _ ~ S . tan x . t 48.siu 7 t 48 . tan 7J t 1 converting both members into time and taking one half, we have, after writing d for #, and making t t = \ X j 1 j ^ P = j*o ^ ^> 5 . tan / . < S . tan rf . t ' 1440 . sin 71 * 1440 . tan 7 * as in the text, page 187. w APPENDIX XIII. CORRECTION FOR DIFFERENCES OF REFRACTION. Let P be the pole, Z the zenith, and S Fi - ia the place of the sun had the air undergone no change, and S' the place as determined by a change of atmospheric refraction. Then, employing the same notation as in the pre- ceding appendix and resuming its equation (a), regarding the altitude a as referring to the place S and P to the hour angle Z P Sj we have, writing d for a?, sin a = sin / . sin d + cos J . cos d . cos P, and denoting the altitude of S' by a' and the hour angle Z P S' by P 1 sin a' = sin I . sin c? + cos I . cos ef . cos P' \ and by subtraction, sin a' sin a = cos / . cos d (cos P / cos P) ; 426 SPHERICAL ASTRONOMY. but fiin a' sin a =2 sin i (a' a) . cos \ (a' + a), cos P' - cos P = 2 sin-J (P - P') . sin 4 (P' + P) ; whence by substitution, sin (a' a) . cos J (a'-f a) = cos I . cos d . sin 1 (P P') . sin \ (P'-f P), and because a and a' as also P' and P differ by very small quantities, the above becomes, by transposing and dividing, p _ p' - K - <*) cos a cos / . cos d . sin P* But denoting the refraction in the afternoon by r' and that in the morning by r, we have a! a = r' r ; substituting and converting both members into time, and writing t tt for the first member, we have ' i (/ - r) . cos a "- Tir 'cos/.cosd.8in.P' *s in the text at page 188. TABLES. TABLE I. Afr. Ivory's Mean Refractions ; with ike Logarithms and their. Differ- ences annexed. i Zenith Dist Mean Refraction. Log. Diff. Zenith Dist Mean Refraction. Log. Diff. o > o i n I 1-02 o.oo85 3oi2 25 o 27.24 1-4352 201 2 2.04 0.3097 i 7 63 26 28-49 i-454 7 i 9 5 - Q.. 3 3.06 0.4860 1252 27 29.76 i-4 7 36 109 o c 4 4-08 0-6112 974 28 3i-o5 1.4921 i85 5 5- ii 0.7086 796 29 32.38 I -5lO2 1 6 6-i4 0.7882 6 7 5 3o 33.72 I .5279 177 7 7.17 o.855 7 58 7 3i 35.09 1.5453 17 8 8.21 0.9144 5i 9 32 36-49 1.5622 170 -CO 9 9.25 o. 9 663 466 33 37-93 1.5790 IOO C. I 10 io3o i .0129 424 34 3 9 .3 9 i.5 9 54 164 _ /?_ ii 11-35 i-o553 388 35 40.89 i6n6 IO2 12 i3 12.42 1-0941 i i3oo 35 9 334 36 3 7 42.42 44-oo i .62-76 1.6435 1 60 i5 9 . c/; i4 14-56 i-i634 3i3 38 45.6i i .6591 1 56 e c i5 i5-66 1-1947 294 39 47-27 i .6746 1 55 16 16-75 1-2241 278 4o 48. 99 1.6901 1 55 .r / 17 17-86 i -2519 265 4i 5o. 7 5 i .7055 ID4 18 18:98 i- 2784 252 42 52.57 I 72OT i5a '9 20- 1 1 i.3o36 241 43 54-43 1.7358 i5i 20 21-26 1.3277 230 44 56-35 i. 7 5io r - 21 22.42 1.3507 222 45 58-36 1.76611 IDI 22 23-6o 1.3729 2l5 46 i o43 1-78123 l5l2 23 24*80 i.3 9 44 207 47 2.5 7 1.79637 i5i4 M o 26*01 i.4i5i 48 i 4-8o r.8u55 < 428 SPHERICAL ASTRONOMY TABLE L (Continued.) Zenith Dist Mean liefruelioii. Log. Diflf. Zenith Dist Mean Refraction. Log. Diff. ' / a o ' / ; 49 o I 7-II 1-82678 i523 7 2 3o 3 3-23 2-26299 429 5o o 9-52 1-84208 i53o 4o 5-o6 2-26 7 32 433 5i o I2-O2 1-85747 i53 9 5o 6-93 2-2 7 l68 436 52 O 14-64 1.87298 i55i 7 3 oo 8-83 2-2 7 6o8 44o 53 o 17-38 1.88863 1 565 IO 10-77 2-28o5l 443 54 o 20-24 i . 90440 1 577 20 I2- 7 4 2-28498 447 55 o 23-25 1-92036 1 5 9 6 3o i4-?5 2-28948 45o 56 o 26-41 i- 9 3653 1617 4o 16-80 2-29402 454 5 7 o 29-73 1-95291 1 638 5o 18-88 2-29860 458 58 o 33-23 1-96955 1 664 7 4 oo 21 -OI 2-3o322 462 5 9 o 36-93 1.98646 1691 IO 2 3-i8 2 -30 7 89 46 7 60 o 4o-85 2-oo368 1722 20 25-3 9 2-31259 470 61 o 45-01 2-02124 i 7 56 3o 2 7 -66 2-3i 7 34 4 7 5 62 o 49-44 2-03918 i 79 4 4o 2 9 . 9 5 2-32213 4 7 9 63 o 54-17 2-05754 1 836 5o 3 2 -3o 2-32696 483 64 o 59-22 2-o 7 635 1881 7 5 oo 34 -70 2-33i84 488 65 o 2 4-65 2-09567 1932 10 3 7 -i6 2-336 77 493 66 o io-48 2-n555 1988 20 3 9 -65 2-34i 7 4 497 67 o i6- 7 8 2-i36o3 2048 3o 42-21 2-346-76 502 68 o 2 3-6i 2-15719 2116 4o 44-82 2-35i83 5o 7 69 o 3i-o4 2-17910 2191 5o 4 7 -48 2-356 9 5 5l2 70 oo 3 9 -i6 2-20185 22 7 5 7 6 oo 5o-2I 2-36212 5i 7 IO 4o-5 9 2-20573 388 10 53-00 2 -36 7 35 523 20 42-04 2 20063 3 9 o 20 55-85 2-3 72 63 528 3o 43-52 2- 2 i 356 3 9 3 3o 58- 7 6 2-3 77 96 533 4o 45-02 2-21752 3 9 6 4o 4 'i- 7 4 2-38334 538 5o 46-53 2'22l5o 3 9 8 5o 4- 7 9 2.388-79 545 71 oo 48- 08 2.22552 402 77 00 7-91 2.39430 55i 10 49-65 2.22956 4o4 IO ii -ii 3.39987 55 7 20 5i.25 2.23363 4o 7 2.O 14-39 2-4o5fo 563 3o 52-87 2-23773 4io 3o i 7 - 7 4 2-41119 56 9 4o 54-53 2. 24l86 4i3 4o 21 -19 2.41695 5 7 6 5o 56.21 2.24603 4i 7 5o 24--2 2.422-8 583 72 oo 57.92 2-25O22 419 7 8 oo 28-0-3 2.42867 58 9 IO 59-66 2-25445 423 10 3 2 .o4 2-43463 596 20 3 1-43 2-25870 ' 425 20 4 35-84 2- 44066 6o3 1 TABLES. TABLE I. (Continued.) Zenith Dist Mean Refraction Log. Difl. Zonith Dist Mean Retraction. Log. Diff. ' 78 3o 4 3 9 - 7 5 2-44677 611 o ' 84 20 8 55-25 2-72856 1069 4o 43.76 2-45295 618 3o 9 8-88 2- 7 3o43 1092 5o 47-88 2.45921 626 4 23-16 2. 7 5o63 iu5 79 oo 52-12 2-46556 635 5o 38-12 2-76202 r,3 9 10 56.47 2-47198 642 85 oo | 53-84 2-773^7 n65 20 5 0-94 2-47848 65o JO ic io-35 2- 7 8558 1 191 3o 5-54 2-48507 659 20 27-73 2-79777 1219 4o 10-28 2-49176 669 3o 46- o3 2-81025 1248 5o i5-i6 2-49853 677 40 ii 5-3o 2-82302 1277 80 oo 20-19 2-5o54i 688 5o 25-66 2-836i i 1 309 10 25-36 2-5l23 7 696 86 oo 47-i5 2-84951 1 34o 20 30-70 2-5i 9 44 707 10 12 9-88 2-86325 i3 7 4 3o 36-20 2.52660 716 90 33-97 2.87735 i4io 4o 4i-88 2-5338 7 727 - 3o 59- 5i 2-89182 1 447 5o 47'74 2-54i25 7 38 4o i3 26-61 2-90666 1 484 81 oo 53-79 2-548 7 4 749 5o i 3 55-4o 2 92 1 89 1 5 2 3 10 6 0-04 2-55635 761 87 oo i4 26-04 2.93754 1 565 20 6-5o 2-56407 772 10 i4 58-71 2-95362 1608 3o i3-i8 2-57192 7 85 20 i5 33-6o 2-97016 1 654 4o 20-09 2.57989 797 3o 16 10-89 2-98717 1701 \ 5o 27-26 2-588oo 811 4o 16 5o-8 3-oo466 1749 82 oo 34-68 2-59624 824 5o 1-7 33-6 3-02267 1801 IO 42-3 7 2-60462 838 88 oo 18 19-6 3-04I22 i855 20 5o-33 2-6i3i3 85i 10 19 9-0 3-o6o3i i 9 o 9 3o 58-59 2-62179 866 20 2O 2-2 3-07998 1967 4o 7 7-19 2-63o62 883 3o 2O 59'6 3- 10024 2026 5o i6-i3 2-63961 899 4o 22 1-7 3-i2ii3 2089 83 oo 25-4o 2-64875 914 5o 23 8-9 3-14268 2i55 10 35-o5 2-658o6 9 3i 89 oo 24 21-8 3-16489 2221 20 45-10 2- 66 7 55 949 IO a5 4o'9 3-18779 220X> 3o 55-58 2-67722 967 20 27 7-1 236l 4o 8 6-5o 2-68708 986 3o 28 4o-8 3-a3574 2434 5o 17-90 2-69714 1006 4o 30 23-2 3.26o83 2509 84 oo 29-80 2-70740 1026 5o 32 i5-o 3-28667 2584 10 8 42-24 2-71787 io47 90 oo 34 17-5 3-5i334 2667 SPHERICAL ASTRONOMY. TABLE II. ifr. Ivonfs Refractions continued : showing the logarithms of the correc- tions, on account of the state of the Thermometer and Barometer. Thermometer. Barometer. Logarithm. Logarithm. Logarithm. o in. 80 9.97237 5o O-OOOOO 3i-o O-OI424 79 9.97326 49 0.00094 3o9 0.01248 78 9.97416 48 0-00190 8 O-O1143 77 9.97506 47 O-OO285 7 O-OIOO2 76 9.97596 46 ooo38o 6 0-00860 75 9.97686 45 0-00476 5 0-00718 74 9.97777 44 o 00572 4 0-00575 73 9.97867 43 0-00668 3 0-00432 72 9-97958 42 0-00764 2 0-00289 7i 9.98049 4i 0-00861 I 000145 70 9.98140 4o 0-00957 3o'0 0-00000 69 9.98231 3 9 ooio53 29-9 9. 99 855 68 9.98323 38 ooii5i 8 9-99709 67 9.98414 37 0-01248 7 9 . 99 563 66 9.98506 36 o-oc346 6 9.99417 65 9.98598 35 o-oi444 5 9.99270 64 9.98690 34 ooi54i 4 9.99123 63 9.98783 33 0-01640 3 9.98975 62 9.98875 32 0-01738 2 9.98826 61 9.98969 3i 0-01837 I 9-98677 60 9.99061 3o 0-01935 29-0 9.98528 5 9 9- % 99'54 29 o-o2o33 28-9 9 . 9 83 7 8 58 9.99248 28 oo2i33 8 9-98227 57 9-9934i 27 0O2232 7 9-98076 56 9.99434 26 - -0233i 6 9-97924 55 9.99529 25 0-02432 5 9.97772 54 9.99623 24 o-o253i 4 9.97620 53 9.99717 23 o-O263o 3 9.97466 52 9.99811 22 0-02730 2 9-973i3 5i 9.99906 21 0-02832 I 9.97158 5o O'OOOOO 20 0-02933 28-0 9.97004 TABLES. 4:31 TABLE III. Mr. Ivortfs Refractions continued : showing the further quantities by which the refraction at low altitudes is to be corrected, on account of the state of the Thermometer and Barometer. Zenith Distance. T B Zenith Distance. T B o ' o ' 7 5 o 0*009 86 3o o3i7 + o-5i 76 o 0*012 86 4o 0-345 o-56 77 o o-oi5 86 5o 0-376 0-62 78 o 0-018 87 o o4io 0-68 79 O'O23 tf 87 10 0.448 0-75 80 o o-o3o + o-o4 87 20 0*490 o-83 81 o o-o4o oo5 87 3o 0-538 0-91 8r 3o o-o46 0*07 87 4o 0.593 IOI 82 o o-o53 0-08 87 5o 0-654 i-i3 82 3o o-o63 OIO 88 o 0.722 1-26 83 o 0-074 O'll 88 10 0-799 i-4i 83 3o 0.089 o.i3 88 20 ,0-887 i-5 9 84 o 0-107 0.16 88 3o 0-987 1-79 84 3o o i3o O2O 88 4o I-IOI 2-02 85 o O'i59 0-25 88 5o I-23l 2.2 9 85 10 0.171 O26 89 o I.38o 2.61 85 20 0-184 0-28 89 10 i-55i 2. 9 8 85 3o 0.198 o.3i 89 20 1-749 3-4i 85 4o 0-2l3 0-33 89 3o 1-977 3. 9 3 85 5o 0-229 0-36 89 4o 2-24l 4-54 86 o 0-248 o-3 9 89 5o 2-549 5-26 86 10 0.269 0-43 90 o -2-909 + 6-12 86 20 0*292 + 0.47 The column marked T is to be multiplied by (t 50); and the column marked B is to be multiplied by (b 3o in -oo). The results are to be applied to the ap- proximate refraction obtained by the -preceding tables. 432 SPHERICAL ASTRONOMY. TABLE IV. For the Equation of Equal Altitudes of the Sun. I nter va Log. A. Log. B. Interval Log. A. Log. B. Interval Log. A. Log. U h. in. h. m. h. m. 2 O 7.7297 7.7146 3 7-735 9 7 - 7 oi5 4 o 7 . 7 44 7 7 .6823 2 .7298 .7143 2 7 362 7 OIO 2 7 45i 68i5 4 73oo 7139 4 7364 7 oo5 4 7 454 6807 6 73o2 7 i36 6 . 7 36 7 6999 6 7 458 6800 8 7 3o4 7 l32 8 7 36 9 6993 8 . 7 46i 6792 10 7 3o5 7128 10 7 3 7 2 6988 10 7 464 >6 7 84 12 7 3o 7 7 I25 12 7 3 7 4 .6982 12 7 468 6 77 6 i4 .7309 7 I2I i4 7377 .6976 i4 7 4 7 2 6 7 68 16 . 7 3n 7117 16 - 7 38o 6970 16 7 4 7 5 6 7 5 9 18 7 3i3 7 Tl3 18 7 383 .6964 18 74/9 6 7 5i 20 . 7 3i5 . 7 io9 20 7 386 6958 20 .7482 6 7 43 22 . 7 3r 7 . 7 io5 22 7 388 6952 22 7 486 6 7 34 24 73i9 7 10l 24 7391 6946 24 7490 6 7 a6 26 7 32I 709-7 26 7 3 9 4 6940 26 . 7 4 9 4 6 7 i 7 ; 28 7 3 2 3 7092 28 739-7 .6 9 34 28 7497 6708 3o . 7 3 2 5 7088 3o 7 4oo .692-7 3o 7 5oi 6 7 oo 32 7 32 7 7083 32 . 7 4o3 .6921 32 7 5o5 6691 34 7329 7079 34 . 7 4o6 -6914 34 7 5o9 .6682 36 . 7 33i 7075 36 7 4o9 6908 36 7 5i3 66 7 3 38 7 333 7070 38 7 4l2 6901 38 75i 7 6663 4o 7 336 7065 4o 7 4i5 .6894 4o . 7 52I 6654 42 7 338 7061 42 7 4i8 .6888 42 7 525 6645 44 . 7 34o 7066 44 7 42I 6881 44 . 7 52 9 6635 46 7 342 7o5i 46 . 7 424 68 7 4 46 7 533 6626 48 7345 7046 48 .7428 .686-7 48 7 53 7 6616 5o 7 34 7 7041 5o 743 1 .6859 5o 7 54i .6606 52 7349 .7036 52 - 7 434 .6852 52 7 545 65 97 54 7 352 7o3i 54 743 7 .6845 54 7 549 .658 7 56 7354 7026 56 r?44i .6838 56 7 553 65 77 58 7.7357 7.7021 58 7 - 7 444 7 -683o 58 7 .755 7 7-6567 . i TABLES. 433 TABLE IV. (Continued.) For the Equation of Equal Altitudes of the Sun. Interval Log. A. Log.B. Interva Log. A. Log. B. Interval Log. A. Log. B. h. m. h. m. b. m. 5 o 7.7562 7.6556 6 o 7-7703 7.6198 7 o 7.7873 7.5717 2 7 566 .6546 2 .7708 6184 2 7879 .56 99 4 .7570 .6536 4 -77*3 6170 4 7 885 568o 6 75 7 5 6525 6 .7719 6i56 6 .7891 566 1 8 7579 65i4 8 .7724 6142 8 .7898 564 1 10 7 583 65o4 10 .7729 6127 10 .7904 6622 12 7 588 6493 12 . 77 35 6n3 12 .7910 .5602 i4 .7692 6482 i4 . 77 4o .6098 i4 .7916 5582 16 7597 -6471 16 7745 6o83 16 .7923 .5562 18 7 6oi 646o 18 775 1 6068 18 .7929 5542 20 7 6o6 6448 20 7756 6o53 20 7936 5522 22 . 7 6io .643 7 22 . 77 62 6o38 22 .7942 55oi 24 7 6i5 6425 24 7767 6023 24 7949 .5480 26 7620 .64:4 26 7773 -6007 26 79 55 545 9 28 7624 6402 28 7779 5991 28 7962 543 7 3o 7629 .63 9 o 3o 7784 .5 97 5 3o 7969 54i6 32 7 634 63 7 8 32 .7790 5 9 5 9 32 .7975 53 9 4 34 7 638 .6366 34 .7796 5 9 43 34 .7982 .53 7 2 36 7 643 .6354 36 .7801 5927 36 .7989 .535o 38 7 648 6342 38 .7807 5910 38 7995 5327 4o 7 653 .632 9 4o . 7 8i3 58 9 4 4o .8002 .53o4 42 7 658 .63i7 42 .7819 .58 77 42 8009 .5281 44 7 663 -63o4 44 .7825 586o 44 .8016 .5258 46 .7668 6291 46 783 1 5843 46 8023 .5234 48 7673 .6278 48 7836 5825 48 8o3o .5211 5o .7678 .6265 9 5o .7842 58o8 5o 8o3 7 5i86 52 - 7 683 -6252 52 7 848 5790 52 8o44 5i62 54 .7688 .6^39 54 . 7 854 5 77 2 54 8o5i 5i3 7 56 .7693 6225 56 7860 5 7 54 56 8o58 5lI2 58 7.7698 7.6212 58 7.7867 7 .5 7 36 58 7 -8o65 7.5087 9.S 434 SPHERICAL ASTRONOMY. TABLE IV. (Continued.) For the Equation of Equal Altitudes of the Sun. Interval Log. A. Log. B. Interval Log. A. Log. B. Interval Log. A. Log.B. h. m h. m. h. m. 8 o 7.8072 7.5062 9 7-8302 7 -4i3i IO 7 .856 7 7 . 2 6 97 2 8079 5o36 2 83n 4o 9 3 2 85 7 6 2635 4 8086 5oio 4 83j 9 4o55 4 8586 25 7 2 6 8094 .4983 6 83 2 8 4oi6 6 8595 .250 7 8 8101 4957 8 8336 .3977 8 86o5 .2442 10 8108 .493o 10 8344 .8 9 3 7 10 86i4 23 7 4 12 8116 4902 12 .8353 .38o6 12 8624 23o6 i4 8i23 .4874 i4 .836 1 3855 i4 .8634 2236 16 8i3o 4846 16 83 7 o 38i3 16 .8643 2164 18 8i38 .4818 18 83 7 8 .3771 18 8653 .2091 20 8i45 .4789 20 838 7 .3728 20 .8663 .20,6 22 8i53 .4760 22 .8396 .3684 22 86 7 3 ip4o 24 8160 473i 24 84o4 3639 24 8683 .1861 26 .8168 4701 26 .84x3 35 9 4 26 8693 .1761 28 .8176 4671 28 8422 3548 28 8 7 o3 .1699 3o .8i83 464o 3o 843o 35oi 3o .8 7 i3 i6i5 32 8191 4609 32 -8439 .3454 32 8 7 23 1529 34 .8199 45 7 8 34 .8448 .34o6 34 8 7 33 .i44o 36 8206 4546 36 845 7 335 7 36 8 7 43 .1349 38 8214 45i4 38 8466 .3307 38 8 7 53 1256 4o 8222 .4482 4c 84 7 5 .3256 4o 8 7 63 .1160 42 823o 4449 42 .8484 .32o5 42 8 77 3 .1061 44 .8238 44i5 44 .8493 3i52 44 .8 7 84 0960 46 8246 438 1 46 85o2 .3o 99 46 8 79 4 o855 48 .8254 4347 . 4 8 .85n 3o45 48 88o4 07,48 5o .8262 43i2 5o .8520 .2989 5o 88i5 o63 7 52 8270 .4277 52 .853o 2 9 33 ' 52 8825 0522 54 8278 .4241 54 .8539 .2876 54 .8836 o4o4 56 8286 .4205 56 8548 .2817 . 56 .8846 0282 58 7.8294 7.4168 58 7 .8558 7 .2 7 58 58 7 .885 7 7 -oi56 TABLES. 435 TABLE IV. (Continued.) For the Equation of Equal Altitudes of the Sun. 1 Interval Log. A. Log. B. Interval Log. A. Log. B. Intcrva' Log. A. Log. B. i b. in h. in. h. m. II O 7-8868 7.0025 12 7.9208 = i3 o 7.9593 7-0750 2 8878 6-9889 2 .9220 -5-5549 2 -9607 0905 4 .8889 9748 4 .9232 5-8641 4 -9620 I0 r ;6 6 8900 .9602 6 .9M5 6-o4r4 6 - 9 634 I2O3 8 8911 9 44 9 8 .9257 .1675 8 9648 i345 10 .8922 .9290 10 .9269 2657 10 .9662 i484 \ 12 .8932 9126 12 .9281 346i 12 .9676 1619 i4 8 9 43 .8953 i4 .9294 4i42 14 .9690 i?5i | 16 8 9 54 .8770 16 - 9 3o6 4?34 16 .9704 1880 18 .8965 858o 18 .9319 .5258 18 .9718 2006 20 .8977 83 79 20 . 9 33 1 .5728 20 - 97 32 .2129 22 .8988 .8168 22 . 9 344 6i54 22 9746 .2249 24 8 999 . 79 45 24 . 9 35 7 .6545 24 .9761 .2367 26 .9010 .7709 26 . 9 36 9 6oo5 26 .9775 2482 28 9021 7 45 7 28 .9382 .7239 28 .9789 2595 3o . 9 o33 .7189 3o 9 3 9 5 755i 3o .9804 2706 32 944 6901 32 .9408 . 7 843 32 .9818 .2815 34 . 9 o55 6591 34 .9421 8119 34 9 833 .2922 36 .9067 6255 36 . 9 433 , .838o 36 9 848 -3o26 38 .9078 .5889 38 9446 .8627 38 .9862 .3i2 9 | 4o 9090 5487 4o .9460 8863 4o -9877 .323i 42 9102 5o4i 47 9473 .9087 42 -9892 .333o 1 44 9ii3 454 1 44 -9486 9302 44 .9907 .3428 46 9125 3 97 3 46 . 9 4 99 9 5 7 46 .9922 .3524 48 9 l3 7 -33i6 48 .9512 .9705 48 . 99 3 7 -3619 5o .9148 .2536 5o .9526 6.9895 5o - 99 52 .3 7 ia 52 9160 1579 52 . 9 53 9 7.0078 52 -9967 .38o4 i 54 .9172 6-o34i 54 9552 .0254 54 -9982 .38 9 4 56 '91 84 5.85 9 3 56 9 566 .0425 56 7.9998 -3o84 58 7-9196 5-55 9 4 58 7 . 9 58o -7-0590 58 8-ooi3 - 7-4071 SPHERICAL ASTRONOMY. TABLE IV. (Continued.) For the Equation of Equal Altitudes of the Sun. Interval Log. A. Log. B. Interval Log. A. Log. B. Interval Log. A. Log. B. I h. m. h. m. h. m. 14 o 8-oo-iS - 7 -4i58 ,5o 8-o52i - 7-635o 16 8-1082 7.8072 2 oo44 4244 2 .o53 9 -64i3 2 IIO2 .8125 4 0059 .4328 4 o556 64 7 5 4 -1122 .8177 6 0075 44i2 6 o5 7 4 .653 7 6 .1143 .8229 8 0090 .4494 8 0592 65 99 8 n63 8abi ; 10 0106 45 7 5 10 0610 6660 10 ii83 .8333 12 OI22 4655 12 0628 6721 12 1204 8385 ! i4 oi38 4 7 35 i4 0646 6781 i.4 1224 8436 16 oi54 48i3 16 .0664 6841 16 1245 848 7 18 0170 .4890 18 .0682 .6900 18 .1266 .8538 20 0186 .4967 20 0700 .6959 20 1287 .8589 22 02O2 .5o43 22 .0718 .7018 22 i3o8 864o 24 02l8 .5n8 24 .0737 .7077 24 1329 8690 26 0234 5192 26 o 7 55 7 i35 26 i35o 8740 28 O25O 5265 28 .0774 .7192 28 -i3 7 i .8790 3o ,0267 5338 3o 0792 .7249 3o i3 9 3 884o 32 0283 54io 32 0811 7 3o6 32 i4i4 8890 34 o3oo 548 1 34 .o83o 7 363 34 i 436 .8 9 3 9 36 o3i6 .555i 36 0849 .7419 36 i 458 8989 38 o333 .5621 38 0868 7475 38 i 479 9038 4o o35o .5690 4o 0887 753i 4o i5or .9087 42 o367 5 7 5 9 42 0906 . 7 586 42 i5tt3 9i36 44 o384 .5827 44 0925 .7641 44 .i545 . 9 i85 46 o4oo . .68.94 46 0945 7696 46 i 568 .9234 48 0417 .5961 48 0964 77 5 1 48 i5 9 o .9282 5o o435 6027 5o .0983 7 8o5 5o 1612 9 33o 52 o452 .6092 52 ioo3 . 7 85 9 52 i635 . 9 3 79 54 0469 6i58 54 1023 .7912 54 i 658 .9427 56 o486 .6222 56 1042 .7966 56 1680 .9475 58 8-o5o4 -7.6286 58 8-1062 7.8019 58 8- 1703 - 7-9523 1 1 TABLES. TABLE IV. (Continued.) for the Equation of Equal Altitudes of the Sun. r* Interval Log. A. Log.B. Interval Log. A. Log. B. Interval Log. A. Log. B. b. m. h. m. b. m. 17 8-1726 -7.9571 18 8-24 7 4 -8.0-969 19 O 8.3359 -8-2354 2 1749 .9618 2 .2501 ioi5 2 33 9 2 2401 4 1773 .9666 4 .2529 1061 4 .3^24 .2448 6 1796 '97l3 6 .2556 1107 6 345 7 .2495 8 1819 .9761 8 .2583 .ii53 8 ,3490 2542 10 .1843 .9808 10 .2611 .1199 10 3524 .2589 12 1867 .9855 12 -2639 .1245 12 355 7 263 7 i4 1890 .9902 i4 .2667 . 1291 i4 .35 9 l -2684 16 -1914 . 99 4 9 16 .2695 i 336 16 .3625 2732 18 .i 9 38 7-9996 18 .2723 .1382 18 3659 .2779 20 .io63 8-oo43 20 .2752 1428 20 -36 9 4 2827 22 .1987 .0090 22 .2781 i474 22 .3728 2875 24 .2011 0137 24 .2809 l52O 24 -3 7 63 2923 26 .2036 .0184 26 .2838 i 566 26 .3 79 8 2971 28 .2061 .0230 28 . 2 68 1612 28 3834 .3019 3o .2086 .0277 3o .2897 -i 658 3o -386 9 3o68 32 .2111 o323 32 .2926 -1704 32 -3 9 o5 .3xx6 34 2i36 .0370 34 - 2 9 56 1750 34 -3 9 4i -3x65 36 2161 0416 36 .2986 -1797 36 3 97 8 .32x4 38 .2186 0462 38 .3oi6 1842 38 /4 ox5 .3263 4o 2212 o5o8 4o 3o46 1889 4o -4o52 .33x2 42 223 7 o555 42 .3077 x 9 35 42 .4089 336 1 44 .2263 0601 . 44 3i07 .1981 44 .4126 .34io 46 .2289 .0647 46 3x38 2028 46 4i64 .346o 48 2 3i5 .0693 48 -3i6 9 2074 48 .4202 -35xo 5c 234i 0739, 5o 3200 2121 5o 4241 .356o 52 2367 o 7 85 52 3232 2167 52 .4279 .36io 54 2394 o83 1 54 -3263 22l4 54 43x8 .366o 56 -2420 0877 56 .3295 226l 56 435 7 3 7 ix 58 8-2447 8-0923 58 8-3327 8-2307 58 8.4397 -8-3 7 6i SPHERICAL ASTRONOMY. TABLE IV. (Continued.) For the Equation of Equal Altitudes of the Sun. Interval Log. A. Log. B. [uterval Log. A. Log. B. interval Log. A. Log. B. h. m. h. m. h. m. 2O O 8-4437 -8-3812 21 8.58io -8.5466 22 O 8.7711 -8. 7 56o 2 4477 .3863 2 5863 .5527 2 .7789 7643 4 45i8 3 9 i5 4 .5917 .5588 4 7868 7727 6 455 9 .3966 6 5971 565o 6 7948 7 8i3 8 .4600 4oi8 8 6025 .5712 8 8o3o .7899 10 .464 1 .4070 10 .6081 5 77 5 10 8n3 .7987 12 4683 4l22 12 6i36 .5838 12 8198 8076 M .4726 4175 i4 6193 .5902 i4 8284 .8167 16 .4768 .4227 16 625o .5 9 66 16 83 7 2 .8259 18 48n .4280 18 63o8 .6o3i 18 846 1 .8353 20 4854 .4334 20 .6366 .6096 20 .8553 .8448 22 4898 438 7 22 .6426 .6162 22 8645 .8545 24 4942 444i 24 .6486 .6229 24 8 7 4o .8644 26 .4987 .4495 26 .6546 .6296 26 883 7 .8 7 45 ' 28 5o32 4549 28 .6608 .6364 28 8 9 35 .8847 3o .5077 .4604 3o 6670 .6433 3o 9 o36 .8952 32 5i23 4659 32 .6733 .6502 32 '9 l3 9 9o58 34 .5169 .47i4 34 .6796 .6572 34 .9244 .9167 36 52i5 .4770 36 6861 6643 36 .9351 ..9278 38 .5262 .4826 38 .6927 .6 7 i5 38 9 46i .9391 4o 53io .4882 4o .6 99 3 .6788 4o 9574 .9507 42 .535 7 .4 9 3 9 42 .7060 .6860 42 .9689 .9626 44 54o6 .4996 44 .7128 .6 9 34 44 .9807 9747 46 .5455 5c >3 46 .7197 .7009 46 8.9928 .9871 48 55o4 . r .n 48 .7268 .708 5 48 9oo52 8-9999 5o 5554 5i6 9 5o . 7 33 9 .7162 5o 0180 9-0129 52 56o4 .5228 52 74ii .7239 52 o3n .0263 54 5655 .5287 54 -7484 7 3i8 54 .0446 o4oi 56 5706 534t> 56 7558 . 7 3 9 8 56 o585 o543 58 8.5 7 58 -8-54o6 58 8.7634 - 8.7478 58 9.0729 _ 9.0689 TABLES. 439 TABLE IV. (Continued.) For the Equation of Equal Altitudes of the Sun. Interval Log. A. Log. B. Interval Log. A. Log. B. Interval Log. A. Log. B. h. m. h. m. b. m. 23 9-0877 - 9.0889 23 2O 9.2693 9-2677 23 4o 9.5761 - 9-5757 2 -1029 .0 99 5 22 .2922 .2907 42 .6224 6221 4 1187 n55 24 3i62 3i49 44 6742 6 7 3 9 6 i35i .1321 26 .34:6 34o4 46 .7328 7 326 8 1620 1492 28 .3685 .36 7 4 48 8oo3 8001 10 1696 1670 3o .3971 .3962 5o .8801 8800 12 [879 -i855 32 .4276 4268 52 9.9776 9 . 977 5 i4 2069 2047 34 46o4 4597 54 o-io3i o-io3i 16 2268 2248 36 4 9 5 7 .4952 56 0-2798 0-2798 18 9.2476 - 9 .2456 38 9-5341 -9-5336 58 o-58i4 -o.58i4 440 SPHERICAL ASTRONOMY. TABLE V. For the Reduction to the Meridian : showing the value of _ 2 sin 2 * P ~~ sin I 77 ""' Sec. O m l m 2 m gm 4 m 5 m gm 7" /f // // n n r oo 2O 7 .8 ' 17-7 3i*4 49-1 70-7 9 6.2 I l oo 2-0 8-0 17-9 3i-7 49.4 71. 1 96.7 2 o-o 2- I 8-r 18.1 31.9 49.7 71.5 97.1 3 O'O 22 8-2 i83 32-2 5oi 71.9 97-6 4 oo 2-2 8-4 18.5 32-5 5o-4 72-3 98-0 5 0-0 v 2>3 8-5 18-7 32. 7 5o. 7 72.7 98-5 6 O'O 2-4 8-7 18-9 33-0 5i-i 7 3-i 99.0 7 oo 2-4 8-8 19-1 33-3 5i-4 7 3.5 99.4 8 oo 2-5 8-9 I9 .3 33-5 5i. 7 73-9 99.9 9 oo 2-6 9-1 19.5 33-8 52-1 74-3 100-4 10 O.I 2-7 9-2 19.7 34-1 52 4 74.7 100.8 ii OI 2-7 9.4 19.9 34-4 5 2 ' 7 7 5- 1 101 .3 12 O-I 2-8 9-5 20-1 34-6 53i_ 7 5-5 ioi8 i3 O'l 2.9 9-6 2O-3 34-9 53-4* 75-9 102.3 i4 o-i 3-0 9-8 20-5 35-2 53.8 7 6.3 102.7 i5 O-I 3-1 9.9 20-7 35-5 54-1 76.7 103.2 16 O'l 3-1 IO-I 20 -9 35-7 54-5 77.1 io3'7 i? 0-2 3-2 IO2 21- 2 36-0 54-8 77-5 io4 2 18 O-2 3-3 10 4 21-4 36-3 55-1 77-9 io4*6 J 9 O2 3-4 10-5 21-6 36-6 55.5 78.3 io5i 20 O2 3-5 10-7 21-8 36-9 55-8 78.8 io5-6 21 O-2 3-6 i0'8 22. O 3 7 .2 56-2 79.2 106.1 22 0-3 3.7 ii -o 22-3 3 7 -4 56-5 79.6 io6'6 23 0-3 3.8 II -2 22-5 3 7 -7 56' 9 80.0 107.0 24 o.3 3-8 ii. 3 22. 7 38-o 5 7 -3 8o-4 107.5 25 0-3 3. 9 ii. 5 22' 9 38-3 5 7 .6 80-8 108-0 26 0-4 4-o ii. 6 23-1 38-6 58-o 8i.3 io8-5 27 0-4 4-i ii-8 23-4 38-9 58-3 81-7 109-0 28 0.4 4-2 ii. 9 23-6 39-2 58. 7 82.1 109.5 29 0-5 4-3 12- I 23-8 3 9 -5 59-0 82-5 Ho-o TABLES. 441 TABLE V. (Continued.) For the Reduction to the Meridian : showing the value of 2 sin 2 P Sec. o m l m 2 m gm 4 m 5 m 6 m 7 , tl a n M u M ; 3o 0-5 4.4 12-3 24'0 3 9 .8 5 9 .4 83-0 i io4 3r 0-5 4-5 12-4 24-3 4oi 5 9 -8 83-4 110*9 32 0.6 4-6 12-6 24-5 4o-3 60. i 83.8 111.4 33 0-6 4-7 12-8 24.7 4o-6 6o-5 84-2 in .9 34 0-6 4-8 12.9 25-0 4o9 60-8 84-7 II2-4 i 35 0-7 4.9 i3-i 25-2 4l-2 61.2 85-i 112-9 36 0.7 5.o i3.3 25.4 41-5' 61.6 85-5 u3.4 3 7 0-7 5.i i3-4 25.7 4i-8 61 .9 86.0 113.9 38 0.8 5-2 i3-6 25.9 42.1 62.3 86-4 "4-4 I 3 9 0-8 5-3 i3-8 26*2 42-5 62-7 86-8 n49 4o 0-9 5-4 i4-o 26.4 42-8 63.o 87-3 u5-4 1 * 0-9 5-6 i4-i 26-6 43.i 63-4 87.7 115.9 42 I O 5. 7 i4-3 26.9 43.4 63-8 88.1 116-4 43 IO 5-8 i4-5 27.1 43-7 64-2 88-6 116.9 44 II 5.9 i4-7 27.4 44-o 64-5 89-0 117.4 45 I ! 6-0 14.8 27.6 44.3 64-9 8 9 .5 117.9 46 I2 6-i i5.o 27.9 44.6 65-3 89.9 118-4 4? ! 6-2 l5-2 28.1 44.9 65. 7 90.3 118-9 48 i3 6.4 i5-4 28.3 45-2 66.0 90-8 119.5 49 i-3 6-5 jS-6 28.6 45-5 66-4 91.2 I2OO 5o i-4 6-6 i5*8 . 28.8 45-9 66-8 91.7 I2O5 5i i-4 6-7 ,5. g 29.1 46.2 67-2 92-1 121. 52 i5 6-8 16-1 29.4 46-5 67.6 92-6 121*5 53 1.5 7.0 i6.3 29.6 46-8 68.0 93.0 I22*O 54 1.6 7.1 i6-5 29 9 47-i 68-3 9 3-5 122-5 55 1.6 7.2 16.7 3o.i 47 5- 68.7 9 3. 9 123-1 56 1.7 7 .3 ' 3o-4 47-8 69.1 94.4 123-6 57 1.8 7-5 17.1 3o-6 48-i 6 9 .5 9 4-8 124*1 58 i-8 7.6 17-3 3o9 48-4 69.9 9 5-3 124-6 59 1.9 7-7 17-5 3i.i 48-8 7 o.3 95.7 125. I 442 SPHERICAL ASTRONOMY. TABLE V. (Continued.) For the Reduction to the Meridian : showing the value of 2 sin' J P ~~ Sec. gm 9 m 10 m ll m 12 m 13 m 14 m ii H u H u ti o 125-7 159. o 196-3 23 7 -5 282-7 33i.8 384-7 i 126*2 169.6 197-0 238-3 283-5 332-6 385.6 2 126-7 160-2 197.6 239-0 284-2 333-4 386-6 3 127-2 160-8 198-3 239.7 285-0 334-3 38 7 -5 4 127-8 161-4 198.9 240- 4 285-8 335-2 388-4 5 128-3 162-0 199-6 24l-2 286-6 336-0 38 9 .3 6 128-8 162-6 200 3 241-9 287-4 336-9 390-2 7 129.3 [63-2 200-9 242-6 288-2 33 7 . 7 391 .1 8 129.9 i63-8 201-6 243.3 289-0 338-6 392.1 9 i3o-4 164-4 2O2-2 244-1 289-8 339.4 3 9 3-o 10 i3i o i65-o 202.9 244-8 290-6 340-3 3 9 3. 9 ii i3r.5 i65.6 203-6 245.5 291.4 34i-2 3 9 4-8 12 l32-O 166-2 2O4'2 246-3 292-2 342-0 3 9 5-8 i3 i3 2 .6 166-8 204-9 247.0 293-0 342.9 396.7 i4 i33-i 167-4 2o5-6 247-7 2 9 3- 8 343.7 3 97 .6 i5 i33.6 168-0 206.3 248-5 294-6 344-6 3 9 8-6 16 i34- 2 168-6 206- 9 249-2 2 9 5-4 345.5 3 99 -5 17 i34-7 169-2 207.6 249-9 296-2 346-4 4oo-5 18 i353 169-8 208-3 25o-7 297-0 347-2 4oi-4 *9 i35-8 170.4 208.9 2 5i-4 297-8 348-1 402.3 20 i36.3 171*0 209.6 252-2 298-6 349-0 4o3-3 21 i36-9 171.6 210-3 253.0 299-4 349-8 4o4a 22 i3 7 -4 172.2 2II-O 253-6 300-2 35o- 7 4o5.i 23 i38-o 172.9 211 -7 254-4 3oi .0 35i-6 4o6o 24 i38-5 i 7 3-5 212-3 255-1 3oi.8 352.5 407.0 25 i39i 174' i 2l3-O 255-9 3o2-6 353-3 4o8-o 26 i3 9 -6 174 7 2l3. 7 256-6 3o3-5 354-2 408-9 27 i4o-2 i 7 5-3 2i4-4 25 7 -4 3o4-3 355-1 409.9 28 i4o7 175.9 2l5-I 258-1 3o5-i 356-0 4io.8 a 9 i4i-3 176-6 2i5-8 208-9 3o5-9 356-9 4n7 TABLES. 443 TABLE V. (Continued.) For the Reduction to the Meridian : showing the value of A = _ 2 sin 2 1 P sin 1' Sec. gm 9 m 10 m 12 m 13 m 14 m 3o i4i-8 ^77-2 216.4 259-6 3o6- 7 357.7 4l2- 7 3i 142-4 177.8 217-1 260-4 307-5 358-6 4i3.6 32 i43.o 178-4 217-8 261 -i 3o8-4 359.5 4i4-6 33 i43.5 179-0 218-5 261 -9 309-2 36o-4 4i5-5 34 i44-i 179-7 219-2 262-6 3io-o 36i.3 4i6-5 35 i44-6 i8o.3 219.9 263-4 3io-8 362-2 4r 7 -5 36 i45.2 180-9 22O-6 264-1 3n-6 363 . i 418-4 3 7 i458 181-6 221-3 264-9 3i2-5 364-0 419.4 38 i46-3 182-2 222 265-7 3i3.3 364-8 420.3 39 146-9 182-8 222-7 266-4 3i4-i 365-7 421.3 4o i47-5 i83-5 223-4 267-2 3i5-o 366-6 422-2 4i i48o 184-1 224-1 267.9 3i5-8 36 7 -5 423-2 42 i48-6 184-7 224-8 268.7 3i6.6 368-4 424-2 43 149-2 i85-4 225-5 269-5 3i 7 -4 369-3 425.1 44 149-7 186-0 226-2 270-3 3i8-3 370*2 426-1 45 i5o-3 186-6 226-9 271 -o 3i 9 .i 3 7 i-i 42 7 -0 46 1 5o 9 187-3 227-6 271-8 319-9 372-0 428-0 47 i5i5 187.9 228-3 272-6 320-8 372-9 429.0 48 152.0 i88.5 229-0 273.3 3ai.6 3 7 3-8 429.9 49 i52.6 189.2 229.7 274-1 322.4 374-7 43o. 9 5o i53.2 189-8 23o-4 274.9 323-3 3 7 5.6 43i-9 5r i53-8 . 190*5 a3i.i 2 7 5.6 324-1 3 7 6.5 432.8 52 154.4 191-1 2 3i-8 276-4 325-0 3 77 -4 433-8 53 i54-9 191-8 232.-S 277-2 325-8 378.3 434-8 54 i55-5 192.4 233-2 278-0 326-7 3 79 -3 435-8 55 i56-i 193.1 234-0 278-8 327-5 38o-2 436. 7 56 i56- 7 193-7 234-7. 279-5 328-4 38i-i 43 7 -7 57 i5 7 -3 194-4 235-4 280-3 329.2 382-0 438. 7 58 157.8 195-0 236-1 281-1 33o.o 382.9 439-7 5 9 i58-4 195-7 236.8 281-9 33o. 9 383-8 44o-6 444 SPHERICAL ASTRONOMY. TABLE V. (Continued.) For the Reduction to the Meridian : showing the value of 2 sin 2 1 P "ihTT 7 " A = Sec. 15 ra 16 m I7 m 18 m 19 m 20 m 21 m o 44i-6 502-5 56 7 -2 635-9 708-4 784-9 865-3 i 442-6 5o3.5 568-3 63 7 .o 70 9 -7 786-2 866-6 2 443-6 5o4-6 56 9 .4 638-2 710.9 787-5 868-0 3 444-6 5o5-6 5 7 o-5 63 9 -4 712-1 788-8 869.4 4 445-6 5o6. 7 571-6 64o-6 7i3-4 790-1 870.8 5 446-5 5o 7 . 7 572-8 64r-7 714-6 791.4 872-1 6 447-5 5o8-8 573.9 642-9 715.9 792-7 8 7 3-5 7 448-5 5o 9 .8 575-0 644-i 717-1 794-0 874-9 8 449.5 510.9 576.1 645-3 718-4 795.4 876-3 9 45o.5 5ii. 9 577-2 646-5 710-6 796-7 877-6 10 45i-5 5i3.o 5 7 8-4 647-7 720. 9 79 8-o 879-0 ii 452.5 5i4-o 579-5 648.9 722-1 799-3 880-4 12 453-5 5i5.i 58o-6 65o-o 723-4 800.7 881-8 i3 454-5 5i6-i 58i. 7 65i-2 724-6 802-0 883-2 i4 455-5 5i 7 -2 582.9 652-4 7 25- 9 8o3-3 88^-6 i5 456.5 5i8-3 584-o 653-6 727-2 8o4-6 886.0 16 45 7 .5 5i 9 -3 585-1 654-8 728-4 8ob-o 887-4 17 458-5 520-4 586-2 656-0 72 9 -7 807-3 888-8 18 45 9 -5 521-5 58 7 .4 657-2 7 3o- 9 808-6 890-2 '9 46o.5 522-5 588-5 658-4 7 32-2 809-9 891-6 20 46i-5 523-6 58 9 -6 65 9 -6 7 33-5 8il.3 893-0 21 462-5 524-6 590-8 ' 660-8 7 34-7 812-6 894-4 22 463.5 525-7 591-9 662-0 7 36-o 8i3- 9 8 9 5- 8 23 464-5 526-8 5 9 3-o 663-2 737-3 8i5-2 897.2 24 465-5 527-9 5 9 4-2 664-4 7 38.5 816-6 808-6 25 466-5 528-9 595-3 665-6 7 3 9 -8 817.9 9 oo-o 26 46 7 -5 53o-o 596-5 666-8 7 4i-i 819-2 001-4 27 468-5 53r-i 597-6 663-0 7 42-3 820-5 9 028 28 469-5 532-2 598-7 66 9 .2 7 43-6 821.9 9 o4-2 2 9 470-5 533-2 599-9 670-4 744- 9 823-2 . 9 o5-ft TABLES 445 TABLE V. (Continued.) For the Reduction to the Meridian : showing the value of 2 sin 2 i P A __ ^ sin I 7 ' ' Sec. 15 m 16 m I7 m 18 m 19 m 20 21 m 3o 471-5 534-3 601 -o 671-6 7 46.2 824 6 907-0 3i 472 6 535-4 602 2 672-8 747-4 825.9 908-4 32 473-6 536-5 6o3-3 674-1 748-7 827.3 909-8 33 474-6 53 7 .6 6o4-5 675-3 75oo 828-6 911-2 34 475-6 538-7 6o5-6 676-5 7 5i-3 829.9 912-6 35 476-6 53 9 -7 606-8 677-7 752-6 83i-2 914-0 36 477-6 54o-8 607.9 678.9 753-8 832-6 9r5-5 3 7 478-7 54r- 9 609. r 680. i 7 55- 1 833-9 916.9 38 479-7 543-0 610-2 681.3 756-4 835-3 918-3 39 480-7 544-i 611.4 682-6 75 7 -7 836-6 919.7 4o 48i- 7 545-2 612.5 683-8 759-0 838-0 921.1 4r 482-8 546-3 6i3. 7 685-0 760-2 83 9 -3 922.5 42 483-8 547-4 614-8 686-2 761-5 84o- 7 923.9 43 484-8 548-4 616-0 687-4 762-8 842-0 925-3 44 485-8 549-5 617-2 688-7 764-1 843-4 926.8 45 486-9 55o-6 6i8-3 689-9 7 65-4 844-7 928-2 46 487-9 55i-7 619-5 691 - r 766.7 846-1 929-6 47 488-9 552-8 620-6 692.4 768-0 847-5 93i o 48 490-0 553-9 621-8 693-6 769-3 848-9 932.4 49 491-0 555-0 623-0 6 9 4- 8 770-6 85o-2 933.8 5o 492-0 556-1 624.1 696-0 771-9 &5i-6 935.2 5i 493.1 557-2 625-3 697.3 77 3-i 852. 9 9 36-6 52 494-r 558-3 626-5 698-5 774-5 854-3 9 38.i 53 495 2 559.4 627-6 699.7 775-8 855-7 9 3 9 .5 54 496.2 56o-5 628-8 701 -o 777.1 857-1 94o "Q 55 497-2 56i-6 63o-o 7O2 2 778-4 858-4 942.3 56 498.3 562 7 63r-2 703^5 779-7 859-8 943.8 57 499-3 563-9 632-3 704 '7 781-0 861-1 945.2 58 5oo-3 565-0 633-5 705-7 782.3 862-5 946-6 59 5oi-4 566-x 634-7 707- 7 83-6 863-9 948-1 SPHERICAL ASTRONOMY. TABLE V. (Continued.) For the Reduction to the Meridian : showing the value of A 2 siQ2 i P ~ sin l /r ~~' Sec. 22 m 23 m 24 m 25 m 26 27 m 28 m 949-6 io37-8 1129-9 1225.9 i325-9 1429-7 i53 7 -5 i 96 ! o 1039.3 n3[-4 1227.5 i32 7 -6 i43i-4 i539-3 2 962.4 io4o8 n33-o 1229-2 i32 9 .3 i433.2 i54i-f 3 953.8 io423 n34-6 i23o-8 i33i-o i434-9 1542-9 4 955.3 io43-8 u36-2 1232-5 i33 2 . 7 i436-7 i544-8 5 9 56. 7 io45-3 1187.8 1234.1 i334-4 1438-5 i546-6 6 968.2 io46-8 ii3 9 -3 1235-7 i336-i i44o.3 i548-4 7 9 5 9 .6 io48-3 ii4o9 i23 7 -3 i33 7 -8 1442-1 i55o-2 8 961 > i 1049-8 ii42-5 1239-0 i33 9 -5 1443-9 1 552- 1 9 962.5 io5i -3 n44'0 1240- 6 i34i-2 i445-6 :553-9 10 963.9 io52-8 ii45-6 1242-3 1342-9 1447.4 r555:8 ii 965-4 io54-3 1147-2 1243-9 i344-6 1449.2 i55 7 -6 12 966.9 io55-9 ii48-8 1245-6 i346-3 i45i o i55 9 -5 i3 968-3 1067.4 n5o-4 1247-2 i348-o i452.8 i56i-3 14 969.8 io58-9 Il52-0 1248-9 1349-7 i454-5 i563-2 i5 971-2 1060-4 ii53-6 i 2 5o-5 i35r-4 i456-3 i565-o 16 972-7 1062-0 u55-2 1252-2 i353-2 i458-i i566. 9 i? 974-1 io63-5 n56-8 1253-8 1354-9 1459.9 i568- 7 18 9 7 5.5 io65-o n58-3 1255-5 1356-6 i46i-6 i5 7 o.5 '9 977.0 1066- 5 1159-9 1257-1 i358-3 i463-4 1572-4 20 978.5 1068- i 6l.5 T258.8 i36o-i i465-2 i5 7 4-3 21 979.9 1069-6 ii63-i 1260.4 i36i-8 1466-9 1576-1 22 981-4 1071-1 1164-7 1262-1 i363-5 i468- 7 1578-0 23 982.9 1072-6 ii66-3 1263-7 i365. 2 i47<>-5 i5 79 .8 24 984-4 1074-2 1167-9 1265-4 i 367-0 1472-3 i58i. 7 25 985-8 1075-7 1169-5 i 267 o i368-7 i474-o 1583-5 26 987-3 1077-2 1171-1 1268-7 1370-4 1475-9 i585-3 27 988-8 1078-7 1172-7 1270- 3 1372-1 i477'7 1587-2 28 990.3 io8o-3 u 7 4-3 1272- i3 7 3- 9 i479' 5 1589.1 29 991-8 1081-8 1175.9 1273.7 i3 7 5-6 i48r-3 1590-9 i i TABLES. 447 TABLE V. (Continued.) For the Reduction to the Meridian : showing the value of _ 2 sin 2 -* P ~ sin 1" ' Sec. 22 m 23 m 24 m 25 m 26 m 27" 28* . 3o 993-2 io83-3 1177.5 1275.4 i3 7 7-4 i483-i 1592-7 3i 994-7 1084-8 1179.1 1277.1 i3 79 .o 1484-9 1594-6 32 996-2 1086-4 1180.7 1278-8 i38o-8 i486- 7 1596-5 33 997-6 1087-9 1182.3 1280.4 i382-5 i488.5 1598-3 34 999.1 1089.5 ii83- 9 1282-1 1384-2 1490-3 1600-2 35 1000-6 1091 -o n85-5 1283-8 1385-9 1492.1 1602- i 36 i 002- i 1092-6 1187-1 1285-5 i38 7 - 7 1493.9 i 6o4 o 3 7 ioo3-5 1094.1 1188-7 1287-1 1389-4 1495-7 1605-9 38 ioo5-o 1095.7 1190-3 1288-8 1391-2 i497-5 1607.7 3 9 i 006 5 1097-2 1191.9 i 290 5 i3 9 2. 9 1499-3 1609.6 4o 1008-0 1098-8 1193.5 1292-2 i3 9 4-7 i5oi -i i6n.5 4i 1009.4 1100.3 1195.1 1293-8 i3 9 6-4 i5o2- 9 i6i3-3 42 1010.9 1101-9 1196.7 1295- 5 i3 9 8-2 i5o4-7 i6i5-a 43 IOI2-4 1*53.4 1198-3 1297.2 1399.9 .1506-5 1617-1 44 1013.9 iio5o 1199.9 1298-9 i4oi7 i5o8-4 1619-0 45 ioi54 iio6-5 I2or 5 i3oo5 i4o3-4 l5[O-2 1620-8 46 1016-9 1108-1 I2O3I 1302.2 i4o5-2 l5l2O 1622-7 47 1018.4 1109-6 1204-7 i3o3- 9 1406.9 i5i3-8 1624-6 43 1019-9 1III. 2 1206.4 i3o5-6 1408-7 i5i5.6 1626.5 49 1021 -4 1112-7 I2O8O i3o7-3 i4io4 i5i 7 .4 1628-3 5o I022- & ni4-3 1209.6 1 309 o l4l22 1519.2 i63o-2 5i io?4-3 ui5-8 I2II.2 i3io7 i4i3'9 l52I -O i632-i 52 1025-8 1117-4 1212*9 i3i2-4 i4]5-7 l522- 9 1634.0 53 1027.3 1118-9 i2i4-5 i3i4-r i4i7-4 1524-7 i635-9 54 1028-8 1I2O-5 1216- i i3i5- 7 1419-2 i5 2 6-5 i63 7 - 7 55 io3o3 fI22-O 1217.7 i3i 7 -4 1420-9 i528-3 i63 9 -6 56 1:531-8 H23-6 1219.4 1319-1 1422.7 i53o-2 1 64 1 5 5 7 io33-3 II25-I 1221 -0 i 320- 8 1424-4 i532-o 1643-3 58 io34-8 1126-7 1222.6 1322.5 1426-2 i533-8 1645-2 59 io36-3 1128-3 1224-2 1324-2 1427-9 i535-6 1647-1 448 SPHERICAL ASTRONOMY. TABLE V. (Continued.) For the Reduction to the Meridian : showing the value of _ 2 sin 2 i P ~~ sin 1" * Sec. 29 m 30 m 31 m 32 ra 33 m 34 m 35 m 1649-0 I 7 64r6 i884-o 2007-4 2i34-6 2266-6 2400-6 i i65o9 1766-6 1886-0 2009.4 2i36.8 2267-8 2402-9 2 1662.8 1768-5 1888- o 2OI1 -5 9 i 38. 9 2270-0 24o5-2 3 1664.7 1770-5 1890-0 2oi3-6 2l4l I 2272- 2 2407 - 5 4 1666-6 1772-4 1892-1 2016.7 2143.2 2274-5 2409-8 5 1658-5 1774-4 1894-1 2017-8 2145-3 2276-7 2412-0 6 1660-4 1776-3 1896-1 2019-9 2i4 7 -5 2278-9 24i4-3 7 1662-3 1778-3 1898-1 2022-0 2149-7 2281.2 2416- 6 8 1664-2 1780-3 I 900 2 2024- I 2161-8 2283-4 2418-9 9 1666-1 1782-3 I9O2.2 2026-2 2153-9 2285-6 2421 -2 10 1668-0 1784-2 1904.3 2028-3 2 i 56- i 2287-8 2423-5 ii 1669-9 1786-2 1906.3 2o3o-5 2i58-3 2290-0 2425.8 12 1671-9 1788-2 1908-4 2032-5 2i6o-5 2292-3 2428.1 i3 i6 7 3- 8 1790-1 I9I0.4 2o34-6 2162-6 2294.5 243o-4 i4 i6 7 5- 7 1792-1 19I2.4 2o367 2164. 8 2296-8 2432-7 i5 1677-6 1794-1 I9I4-4 2o38.8 2166-9 2299-0 2435-0 16 1679.5 1796-1 1916-6 2040 9 2 1 69 - I 230I 3 2437-3 '7 1681-4 1798-1 1918.5 2o43- o 2I 7 f.2 23o3-6 2439-6 8 i683-3 1800. o 1920-6 2045 i 2173.4 23o5-8 2441-9 '9 i685-2 1802-0 1922.6 2047 ' 2 2176-6 23o8-o 2444-2 20 1687-2 i8o4-o I924-7 2049.3 2177-8 23lO-2 2446-5 21 1689-1 i8o5-9 1926-7 2o5i -4 2179-9 2312-4 2448-8 22 1691 -o 1807-9 1928-8 2053-5 2I82.I 23i4-7 2461-1 23 1692-9 1809.9 1930.8 2055-7 2i84-3 2316-9 2453-4 24 1694-8 1811 -9 i 9 32- 9 2067-8 2186-6 2319-2 2455. 7 25 1696-7 1813.9 1935-0 2069-9 2188.6 2321-5 2458-0 26 1698-6 i8i5-8 1937-0 2062-0 2190.8 2323-7 2460.3 27 1700-6 1817-8 1939-0 2064 i 2193.0 2325-9 2462-6 28 1702-6 1819-8 1941-1 2066-2 2196.2 2328-2 2464-9 29 1704-4 1821-8 i 943 . i 2068.3 2197-3 2467.2 TABLES. 449 TABLE V. (Continued.) For the Reduction to the Meridian : showing the value of 2 sin 2 I P Sec. 29 m 30 m 31 m 32 m 33 m 34 m 35 m 3o 1706-3 1823-8 1945.2 2070-4 2199.5 2332. 7 246 9 .5 3i 1 708 2 1825.8 1947.2 2072-6 22OI 7 2334-9 2471-8 32 I7I0.2 1827-8 1949.3 2074-7 2203.9 233 7 -2 2474.2 33 I7I2.I 1829-8 i 9 5i.3 2076-8 2206.1 2339.4 2476-5 34 I7l4'0 i83i-8 1953.4 2078*9 2208.3 2 34r-7 2478-8 35 I7I5.9 1833-8 i 9 55- 5 2081 -o 221O-5 2343-9 2481-1 36 1717.9 i835-8 1957-6 2083-2 2212-7 2346-2 2483-5 3? 1719-8 i83 7 -8 1959-6 2085-3 2214-9 2348-5 2485-8 38 I72I.7 1839-8 1961-7 2087.4 2217-1 235o. 7 2488.1 3 9 1723.6 i84i-8 i 9 63- 7 2089-6 2219-3 2353-0 2490.4 4r 1725.6 i843-8 I 9 65.8 2091-7 2221 -5 2355-2 2492-8 4: 1727.5 1845-8 1967-8 2093-8 2223-7 2357-5 2495-1 42 1729.5 1847- 8 1969.9 2095-9 2225*9 235 9 - 7 2497.4 43 I73I.5 1849-8 1972-0 2098-0 2228-1 236i -9 249 9 -7 44 1733.4 i85i-8 1974-1 2IOO-2 2230-3 2364-a 25O2I 45 1735-3 1853-8 1976-1 2102-3 2232-5 2366-4 25o44 46 I737.-J 1855-8 1978-2 2io45 2234-7 2368-7 2 5o6. 7 4? 1739.2 1857.8 1980-3 2106. 6 2236-9 2371-0 2509. o 48 I74l-2 1859.8 1982-4 2108-8 2239-1 2 3 7 3.3 25ri-4 49 1 743-. I 1861-8 1984-4 2IIO9 2241 -3 2 3 7 5.5 2 5t3. 7 5o 1745.1 1863-8 1986-5 2Il3 I 2243-5 2 3 77 -8 2 5i6.i 5: 1747-0 i865.8 1988-6 2Il52 2245-7 238o-i 25i.8-4 52 1749-0 1867-8 1990-7 2117.4 2247-9 2382-4 2520-8 53 1750.9 1869.8 1992.7 2119.6 225O-I 2384 6 2523-1- 54 1752-9 1871-8 1994-8 2121.7 2252-3 2386-9 2525-4 55 1754- 8 i8 7 3. 8 1996-9 2123-8 2254-5 238 9 .2 2527.7- 56 i 7 56-8 1875-9 1999.0 2126-0 2256-7 23 9 i-5 253o-i 5 7 i 7 58. 7 1877-9 2001 -0 2 I 28 . I 2258.9 23 9 3- 7 2532.4 58 1760-7 1879.9 2OO3-I 2i3o-3 2261 i 23o6-o 2534.8 5 9 1762.6 1882-0 2005-3 2132-4 2263-4 2 3 9 8-3 2537-1 i 450 SPHERICAL ASTRONOMY. TABLE VI. For the second part of the Reduction to the Meridian : showing the value of 2sin 4 -*P sin 1" * Minutes s 10 20 s 30' 40" 50 s M M a M M // 5 O-OI o.oi O-OI O-OI OOI O-OI 6 OOI OOI O-OI 0.02 0-02 OO2 7 OO2 OO2 oo3 oo3 o-o3 0-04 8 0-04 oo4 o-o5 oo5 o-o5 oo6 9 oo6 0*07 0-08 0-08 0-08 0-09 10 0-09 OIO OII OI I OI2 o-i3 ii o.i4 o-i5 o.i5 oi6 0-17 0-18 12 0-19 O-2O O-22 O.23 0.24 0-25 i3 0-27 0.28 o3o o-3i 0-33 0-34 i4 0-36 o-38 0.39 o-4i 0-43 o-45 i5 0-47 o49 0-52 0-54 o-56 o.5 9 16 O'6i o-64 0-67 0*69 0-72 o. 7 5 i? 0.78 0-81 o-84 0-88 0-91 0-95 18 0-98 I O2 i o6 1.09 i-i3 1.18 '9 I .22 1-26 i-3o 1-35 i4o i-44 20 1.49 1-54 i 6o i.65 1.70 1.76 21 1.82 1.87 i- 9 3 1.99 2.06 2I2 22 2.1 9 2-25 2-32 2 .3 9 2.46 2-54 23 26l 2.69 2. 77 2-85 2. 9 3 3.oi 24 3.10 3.18 3.27 3-36 3.45 3.55 25 3-64 3.74 3-84 3. 9 4 4-o5 4-i5 26 4.26 4-3 7 4-48 4-6o 4.72 4-83 2 7 4.96 5-o8 5 -20 5-33 '5-46 5-6o 28 5. 7 3 5-87 6oi 6-i5 6-3o 6-44 2 9 6.59 6-75 6*90 7.06 7.22 7-38 3o 7-55 7.72 7.89 8-06 8-24 8.42 3i 8-61 8-79 8-98 9.17 9 .3 7 9-57 32 9.77 9.97 10. 18 10.39 io6i 10-82 33 ii -o4 ii .27 ii 5o ii. 7 3 ii .96 I 2 2O 34 12-44 12*69 12-94 13-19 i3-45 i3. 7 i 35 13.97 !4-24 i4-5i 14-78 i5-o6 i5-15 TK1GONOMETKICAL FOUMULJE. I. Equivalent expressions for sin & 1. cos x . tan x. cos x 2. cot x 7. 8. 9. n A /l cos 2 * 2 2 tan x 1 + tan 1 Jar" 2 cot \ x + tan J i sin (30 + a;) - B sin (30 x) 13. VI 11. 2 sin(45 -f i *) 1. 12. 1 2 sin 8 (45 ar). 1 - tan 2 (45 - ^ ar) 1 4- tan 2 (45 - *) * tan (45 + j *) - tan (45 - $ a?) tan (45 + i *) + tan (45 J ar)' 16. sin (60 + a;) sin (60 x). 1 16. oowcant x 452 SPHERICAL ASTRONOMY. D. Equivalent expressions tor cos 1. sin x tan x 2. 3. sin x . cot x. Vl sin 8 x. 4. 1 Vl 4- tan 2 x 5. cot x Vl 4- cot 8 "* 0. cos 2 ^ a; sin 8 as. 7. 1 2 sin 8 Jar. 8. 9. 2 cos' J x - 1. 4 /l -f- cos 2 a: 2 10. 1 tan 2 J x 1 H tan 2 J*' 11. cot J ar tan J x cot J a; 4- tan J a: ' 12. 1 1 4" tan x . tan ^ a? * 13. 2 tan (45 4- } *) 4- cot (45 4- j a?) 14. 2 cos (45 4- J x) cos (45 Jar). 15. cos (60 4- a:) 4- cos (60 *). 16. 1 secant x' TRIGONOMETRICAL FORMULAE 453 HI. Equivalent expressions for tan x. 1. sm x COB X 1 cot a:' 4. sm x 6. 7. 8. Vl cos 8 x COS X 2 tan a; 1 tan* Jar" 2 cot j x cot 8 a; 1 ' 2 cot J * tan f x 9. cot x 2 cot 2 *. 1 cos 2 x 10. 11. sin 2 x sin 2 # + cos 2 or* 19. cos 2 a; 1 4- cos 2 a? tan (45 + *) tan (45' 4.54 SPHERICAL ASTRONOMY. IV. Kelative to two arcs A and B. 1. sin (A 4- B) = sin A . cos B + cos A . sin B. 2. sin (A B) = sin A . cos 2? cos A . sin ./?. 3. cos (A -\- B) = cos A . cos J? sin A . sin J5. 4. cos (^4 -B) = cos A . cos B + sin -4 . sin B. 5 tanU + JS) _ tan ^ + tan ^ tan A tan J5 6. tan ^-5 = ---- 7. sin (45 5) 1 tan 9. tan (45- 1?) ^H sin B 10. 1 db sin B cos .5 11. sin (-4 + B) tan -4 + tan B '__ cot ^ + cot A 12 ' sin (A ^?) = tan A tan ~~ cot B - cot ^4 cos (A + B) cot J? tan A _ cot ^4 tan ^ 13 ' cos (^4 B) ~ cot B -f tan -4 ~~ cot A + tan B ' sin ^. + sin ^ tan ^ (^1 + B) 14. 15. sin A sin tan % (A B) cos jg-f cos^L cot |(^4 + ^g) cos ^ cos J. "~ tan i '^1 B) ' [continued. TRIGONOMETRICAL FORMULAE. 455 IV. continued. Eelative to two arcs A and B. 16. sin A . cos B = sin (A + B) + sin (A B). 17. cos A . sin B = $ sin (A + B) % sin (A B). 18. sin A . sin B = % co* (A B) % cos (A + ^?). 19. cos A . cos 5 = J cos (J. -f B) + i cos (Ji . B). 20. sin J. + sin = 2 sin i (^4 + .5) . cos % (A B). 21. cos A H- cos ^ = 2 cos J (^4 + B) . cos ^ (A ,5). , sin (A + -B) sin ( A + ) 23. cot A + cot ^ =-. 24. sin ^1 - sin B = 2 sin 1 (^ - ^) . cos % (A + B). 25. cos B cos A = 2 sin ^ (A B) . sin i (^ + #) sin (^ - B) 26. sin (A - 27. cot B cot A = - sin -4 . sin B ' 28. sin 2 A sin 2 5 ) V = sin (.4 B) . sin (^4 + B). 29. cos 2 J? cos 2 A ) ' 30. cos 1 A sin 2 B = cos (^4 B) . cos (^4 + B). sin (A ) . sin (A + B) 31. tan' 4 - trf . - - ^.^A- f sin (A B) . sin (A + -5) 456 SPHERICAL ASTRONOMY. V. Differences of trigonometrical lines. 1. A sin a; = -f 2 sin -J A x . cos (a; -f i A a?). 2. A cos a; = 2 sin \ A x . sin (x + % A ar). 3. A tan a: = + sin A * 4. A cot a; = cos x . cos (x + A sin A # sin ar . sin (x + A a;) ' 5. A sin 2 x = + sin A a; . sin (2 x + A x). 6. A cos 2 x = sin A a: . sin (2 # -f A x}. 7. A tan 2 x = + Sin A * 8iD ( 2 * + A ^) cos 2 x . cos 2 (a; -f- A a:) 8. A cot x = - sin 2 x . sin 2 (a? -f A a?) VI. Differentials of trigonometrical lines. 1. d sin x =. -f- d x . cos x. 2. d cos a; rr d x . sin x. da? 8. d tan a; = cos* x ' dx 4. d cot x = - . siir x 5. d sin 2 x = + 2 d a; . sin x . cos ?. 6. d cos 8 x = 2 d a; . sin x . cos jr. 7. d tan'* = + 2d *- tan *. cos 2 a: Q j j g 6 vJ C C\ sin 2 x TRIGONOMETRICAL FORMULAE. 457 VII. General analytical expressions for the sides end angles of any spherical triangle. 1. cos S = cos .4 . sin ' . sin S" + cos S' . cos S" 2. cos S' = COB A' . sin S" . sin S' + cos S" . cos S. 3. cos S" = cos A" . sin S . sin S' + cos S . cos S'. 4. cos ^4 = cos S . sin ./!' . sin A" cos ^4' . cos A". 5. cos A' = cos $' . sin A" . sin ^4 cos A" . cos A. 6. cos A!' = cos 5" . sin A . sin A' cos J. . cos A'. 7. cos .cos ^4' = cot S" . sin S - sin .4' . cot A" 8. cos ' . cos A" = cot S . sm ' sin ^4" . cot ^1 9. cos S" . cos A cot ' . sin /S" sin ^4 . cot A'. sin A _ sin A' _ sin A" sin ~~ sin S f sin S" ' 11. sin \ (S r + 5) : sin }(& S) : : cot 4" : tan (A 1 - ^4). 12. cos (' + 5) : cosi (' _ ^) : : cot J 4" : tan J (^' + ^4). 13. sin J (^4' + ^4) : sin \(A' A) : : tan S" : tan i (S r - ^). 14. cos J (^' + -4) : cos$ (A' - ^4) : : tan S" : tan } (' + ^). In these formulae A, A', A", denote the several angles of the triangle ; and S, S', 5", the sides opposite those angles respectively. For the more convenient computation of the formulas Nos. 1-9, certain auxiliary angles are introduced, which will be alluded to in the formulae for the solution of the several cases of oblique-angled spherical triangles. 458 SPHERICAL ASTRONOMY. VlLL. Solutions of the cases of right-angled spherical triangles. Required. Solution. s ' n x ~ sin h sin Given. Hypothen and < side adj. giv. ang. 2. tan x tan h . cos a. an angle. ,, ,, v. the other angle. 3. cot x = cos h . tan a. Hypothen. and the other side. 4. cos x = cos h cos $' ang. adj. giv. side. 5. cos x = tan s . cot h. sm s ang. op. giv. side. 6. sin x = - sin h the hypothen. the other side, the other angle. *7. sin x = 8. sin x sin s tho ambiguous cases. sin a tan s . cot a cos a cos * A side and the angle < opposite. A side and f tlle tyP otlien - 10 - cot x = cos a . cot s. the angle J the other side. 11. tan x = tan a . sin s. adjacent. ,, ^ I tne other angle. 12. cos x = sm a . cos s. The two ( the hypothen. 13. cos x rectang. cos of the given sides. sides. an The two an g les - 1 a side. the hypothec. 15. cos x = rectang. cot of the giv. angles ., cos opp. 16. cos x = sm ang. In these formulae, x denotes the quantity sought. a = the given angle. s = the given si<'e. A = the hypothenuse. TRIGONOMETRICAL FORMULAE. 4.59 IX. Solutions of the cases of oblique-angled spherical triangles. GIVEN, Two sides and an angle opposite one of them. Required, 1. The angle opposite the other given side. sin side op. ang. sought x sin giv. ang. sm x . . . 8 2-: f - sin side oppos. given angle Required, 2. The angle included between the given sides. cot a! = tan giv. ang. x cos adj. side, cos a' x tan side adi. giv. ang. cos a = : :L - 5 : S-, tan side op. given angle x = (a' a"). Required, 3. The third side. tan a' = cos giv. ang. x tan adj. side, cos a' x cos side op. giv. ang. cos side adj. given angle x = (a' =b a"). In these formulae, x denotes the quantity sought : a f and a" are auxiliary angles introduced for the purpose of facilitating the compu- tations. The angle sought in formula 1 is, in certain cases, ambiguous. In the formulae 2 and 3, when the angles opposite to the given sides are of the same species, we must take the upper sign ; on the contrary, the lower sign. The whole of these formulae therefore are, in certain cases, ambiguous. [continued. 4:60 SPHERICAL ASTRONOMY. IX. continued. Solutions of the cases of oblique-angled spherical triangles. GIVEN, Two angles and a side opposite one of them. Required, 4. The side opposite the other given angle. sin ang. op. side sought X sin giv. side sin x = -2 J s_ __ m sin ang. op. given side Required, 5. The side included between the given angles. tan a' = tan giv. side X cos ang. adj. giv. side, sin a' X tan ang. adj. giv. side sin a = =-r rt ' tan ang. op. given side x = (a 1 =fc a"). Required, 6. The third angle. cot a' = cos given side X tan adj. angle, sin a' X cos ang. op. giv. side cos ang. adj. given side x = (a' a"). In these formulae, x denotes the quantity sought: a' and a" aie auxiliary angles introduced for the purpose of facilitating the compu- tations. The side sought in formula 4 is, in certain cases, ambiguous. In the formulae 5 and 6, when the sides opposite the given angles are of the same species, we must take the upper sign; on the contrary, the lower sign. The whole of these formulae therefore are, in certain cases, ambiguous. [continued. TRIGONOMETRICAL FORMULAE. IX. continued. Solutions of the cases of spherical triangles. GIVEN, Two sides and the included angle. Required, 7. One of the other angles. tan a' = cos given angle X tan given side, a" the base a' sin a' tan x = tan given an^le x - 77. sm a," In this formula, the given side is assumed to be the side opposite the angle sought : the other known side is called the base, Required, 8. The third side. tan a' = cos given angle X tan given side, a" = the base ~- a r , cos a" cos x = cos given side X - r cos a' In this formula, either of the given sides may be as sumed as the base; and the other as the given side. In these formulae, x denotes the quantity sought : a' and a" are auxiliary angles introduced for the purpose of facilitating the compu- tations. If the side sought in formula 8 be small, the formula may not give the value to a sufficient degree of accuracy * and some other mode must be adopted for obtaining the correct value. [continued. 462 SPHERICAL ASTRONOMY. IX. continued. Solutions of the cases of spherical triangles. GIVEN, A side and the two adjacent angles. Required, 9. One of the other sides. cot a' = tan given angle X cos given side, a" = the vertical angle ~ a', cos a' tan x = tan given side X - r. . cos a" In this formula, the angle, opposite the side sought, is assumed as the given angle : the other known angle is called the vertical angle. Required, 10. The third angle. cot a' = tan given angle X cos given side, a" the vertical angle a', sin a" cos x = cos mven anme X - r . sm a' In this formula, either of the given angles may be assumed as the vertical angle ; and the other as the given angle. In these formulae, x denotes the quantity sought: a f and a" are auxiliary angles introduced for tho purpose of facilitating the compu- tation.*. If the angle sought in formula 10 be small, the formula may not give the value to a sufficient degree of accuracy; and some other mode must be adopted for obtaining the correct value [continued. TRIGONOMETRICAL FORMULA. IX. continued. Solutions of the cases of spherical triangles. GIVEN, The three sides. Required, 11. An angle. /A + B + C \ [A + B + C 2 --- B ) X Sm ( -- 2 sin B. sin C sin ,0 . sm In these formulae, -4, 5, C are the three sides of the triangle ; and A is assumed as the bide opposite to the angle required. GIVEN, The three angles. Required, 12. A side. a + b + c >S _ sin b . sin c (a + b+c T \ /a + b + c \ 6 ) xcos ( e ) \_2 sin 6 . sin c In these ^tvtnulee, a, o, c are the three angles of the triangle ; and a is assumed as the angle opposite to the side required. In these formulae, x denotes the quantity sought. The formulae, which are resolved hy the cosine, are used only when the angle or side x is mi all. 464 SPHERICAL ASTRONOMY. X. Trigonometrical series. 2. 2.3.4 2.3.4.5.6 2 a; 5 17 x 1 iTT + F7 sTT + 5. vernam x = 2 2.3.42.3.4.6 sin 3 a; 1 . 3 sin 5 x 7. 2.4.5 cos 3 x 1.3 cos 5 x 8. a; = tan x J tau ? x + J tan 5 ar